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Class. l.i_lA_- 



Book. 



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COPYRIGHT DEPOSIT. 



ENGINEERING 
DESCRIPTIVE GEOMETRY 



Engineering 
Descriptive Geometry 



A Treatise on Descriptive Geometry as the Basis of Mechanical 

Drawing, Explaining Geometrically the Operations 

Customary in the Draughting Room 



fIw. bartlett 

COMMANDER, U. S. NAVY 

HEAD OF DEPARTMENT OF MARINE ENGINEERING AND NAVAL CONSTRUCTION 
AT THE UNITED STATES NAVAL ACADEMY 

AND y.*^ 

THEODORE W; JOHNSON 



PROFESSOR OF MECHANICAL DRAWING, UNITED STATES NAVAL ACADEMY 
MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS 



NEW YORK 
JOHN WILEY & SONS 

London: CHAPMAN & HALL, Limited 
1910 






Copyright, 1910, by 
T. W. Johnson 



Z^t JSorb (§aU\mou (pttef 

BALTIMORE, MD., V. B. A. 



»CI.A27173\ 



PEEFACE. 

The aim of this work is to make Descriptive Geometry an 
integral part of a course in Mechanical or Engineering Drawing. 

The older books on Descriptive Geometry are geometrical rather 
than descriptive. Their authors were interested in the subject as a, 
branch of mathematics, not as a branch of drawing. 

Technical schools should aim to produce engineers rather than 
mathematicians, and the subject is here presented with the idea 
that it may fit naturally in a general course in Mechanical Drawing, 
It should follow that portion of Mechanical Drawing called Line 
Drawing, whose aim is to teach the handling of the drawing instru- 
ments, and should precede courses specializing in the various 
branches of drawing, such as Mechanical, Structural, Architectural, 
and Topographical Drawing, or the " Laying Off " of ship lines. 

The various branches of drawing used in the different industries 
may be regarded as dialects of a common language. A drawing is 
but a written page conveying by the use of lines a mass of informa- 
tion about the geometrical shapes of objects impossible to describe 
in words without tedium and ambiguity. In a broad sense all these 
branches come under the general term Descriptive Geometry. It 
is more usual, however, to speak of them as branches of Engineer- 
ing Drawing, and that term may well be used as the broad label. 

The term Descriptive Geometry will be restricted, therefore, to 
the common geometrical basis or ground work on which the various, 
industrial branches rest. This ground work of mathematical laws, 
is unchanging and permanent. 

The branches of Engineering Drawing have each their own: 
abbreviations and special methods adapting them to their OAvn 
particular fields, and these conventional methods change from time 
to time, keeping pace with changing industrial methods. 

Descriptive Geometry, though unchanged in its principles, has 
recently undergone a complete change in point of view. In 
changing its purpose from a mathematical one to a descriptive one, 
or, from being a training for the geometrical powers of a mathema- 
tician to being a foundation on which to build up a knowledge of 



vi Preface 

some branch of Engineering Drawing, the number and position of 
the planes of projection commonly used are altered. The object is 
now placed behind the planes of projection instead of in front of 
them, a change often spoken of as a change from the " 1st quad- 
rant " to the " 3d quadrant," or from the French to the American 
method. We make this change, regarding the 3d quadrant method 
as the only natural method for American engineers. All the prin- 
ciples of Descriptive Geometry are as true for one method as for the 
other, and the industrial branches, as Mechanical Drawing, Struc- 
tural Drawing, etc., as practiced in this country, all demand this 
method. 

In addition, the older geometries made practically no use of a 
third plane of projection, and we take in this book the further step 
of regarding the use of three planes of projection as the rule, not 
the exception. To meet the common practice in industrial branches, 
we use as our most prominent method of treatment, or tool, the 
auxiliary plane of projection, a device which may be called the 
draftsman's favorite method, but which in books is very little 
noticed. 

As the work is intended for students who are but just taking up 
geometry of three dimensions, in order to inculcate by degrees a 
power of visualizing in space, we begin the subject, not with the 
mathematical point in space but with a solid tangible object shown 
by a perspective drawing. No exact construction is based on the 
perspective drawings which are freely used to make a realistic ap- 
pearance. As soon as the student has grasped the idea of what 
orthographic projection is, knowledge of how to make the projection 
is taught by the constructive process, beginning with the point and 
passing through the line to the plane. To make the subject as 
tangible as possible, the finite straight line and the finite portion of 
a plane take precedence over the infinite line and plane. These 
latter require higher powers of space imagination, and are therefore 
postponed until the student has had time to acquire such powers 
from the more naturally understood branches of the subject. 

F. W. B. 
T. W. J. 

Maech, 1910. 



CONTENTS. 

CHAPTEB PAGE 

I. Nature of Orthographic Projection 1 

II. Orthographic Projection of the Finite Straight Line... 18 

III. The True Length of a Line in Space 27 

IV. Plane Surfaces and Their Intersections and Develop- 

ments 38 

V. Curved Lines 49 

VI. Curved Surfaces and Their Elements G2 

VII. Intersections of Curved Sutifaces 70 

VIII. Intersections of Curved Surfaces; Continued 81 

IX. Development of Curved Surfaces 91 

X. Straight Lines of Unliaiited Length and Their Traces. . 98 

XI. Planes of Unlimited Extent: Their Traces 108 

XII. Various Applications 121 

XIII. The Elements of Isometric Sketching 133 

XIV. Isometric Drawing as an Exact System 142 

Set of Descriptive Drawings 149 



CHAPTER I. 
NATTTIIE OF ORTHOGRAPHIC PROJECTION. 

1. Orthographic Projection. — The object of Mechanical Draw- 
ing is to represent solids with such mathematical accuracy and 
precision that from the drawing alone the object can be built or 
constructed without deviating in the slightest from the intended 
shape. As a consequence the " working drawing " is the ideal 
sought for, and any attempt at artistic or striking effects as in 
" show drawings " must be regarded purely as a side issue of minor 
importance. Indeed mechanical drawing does not even aim to 
give a picture of the object as it appears in nature, but the views 
are drawn for the mind, not the eye. 

The shapes used in machinery are bounded by surfaces of mathe- 
matical regularity, such as planes, cylinders, cones, and surfaces 
of revolution. They are not random surfaces like the surface of a 
lump of putty or other surfaces called " shapeless." These definite 
shapes must be represented on the fiat surface of the paper in an 
unmistakable manner. 

The method chosen is that known as '"'' orthographic projection/" 
If a plane is imagined to be situated in front of an object, and 
from any salient point, an edge or corner, a perpendicular line, 
called a " projector," is drawn to the plane, this line is said to 
project the given point upon the plane, and the foot of this perpen- 
dicular line is called the projection of the given point. If all 
salient points are projected by this method, the orthographic draw- 
ing of the object is formed. 

2. Perspective Drawing. — The views we are accustomed to in 
artistic and photographic representations are " Perspective Views." 
They seek to represent objects exactly as they appear in nature. 
In their case a plane is supposed to be erected between the human 
eye and the object, and the image is formed on the plane by sup- 
posing straight lines drawn from the eye to all salient points of 



2 



Engineerixg Descriptive Geometry 



the object. Where these lines from the eye, or " Visual Eays/' as 
they are called, pierce the plane, the image is formed. 

Fig. 1 represents the two contrasted methods applied to a simple 
object, and the customary^ nomenclature. 

An orthographic view is sometimes called an " Infinite Perspec- 
tive View," as it is the view which could only be seen by an eye at 
an infinite distance from the object. " The Projectors " may then 
be considered as parallel visual rays which meet at infinity, where 
the eye of the observer is imagined to be. 




s:a: 
Perspective View. 




Orthographic View. 



Fig. 1, 



3. The Reg-ular Orthographic Views. — Since solids have three 
" dimensions," length, breadth and thickness, and the plane of the 
paper on which the drawing is made has but two, a single ortho- 
graphic view can express two only of the three dimensions of the 
object, but must always leave one indefinite. Points and lines at 
different distances from the eye are drawn as if lying in the same 
plane. From one view only the mind can imagine them at dif- 
ferent distances by a kind of guess-work. If two views are made 
from, different positions, each view may supplement the other in 
the features in which it is lacking, and so render the representa- 
tion entirely exact. Theoretically two views are always required 
to represent a solid accurately. 

To make a drawing all the more clear, other views are generally 
advisable, and three views may be taken as the average requirement 
for single pieces of machinery. Six regular views are possible, 
however, and an endless number of auxiliary views and " sections " 
in addition. For the present, we shall consider only the " regular 
yiews,** which are six in number. 



Nature of Orthographic Projection 



4. Planes of Projection. — A solid object to be represented is 
supposed to be surrounded by planes at short distances from it, the 
planes being perpendicular to each other. From each point of 
every salient edge of the object, lines are supposed to be drawn 
perpendicular to each of the surrounding planes, and the succes- 
sion of points where these imaginary projecting lines cut the planes 
are supposed to form the lines of the drawings on these planes. 
One of the planes is chosen for the plane of the paper of the actual 
drawing. To bring the others into coincidence with it, so as to 
have all of them on one flat sheet, they are imagined to be unfolded 
from about the object by revolving them about their lines of inter- 
section with each other. These lines of intersection, called " axes 
of projection," separate the flat drawing into different views or 
elevations. 




Fig. 2. 



Engineering Descriptive Geometry 




Fig. 2a. 



Fig. 2 is a true perspective drawing of a solid object and the^ 
planes as they are supposed to surround it. This figure is not a. 
mechanical drawing, but represents the mental process by which 
the mechanical drawing is supposed to be formed by the projection 
of the views on the planes. In this case the planes are supposed 
to be in the form of a perfect cube. The top face of the cube shows 
the drawing on' that face projected from the solid by fine dotted 
lines. Eemember that these fine dotted lines are supposed to be 
perpendicular to the top plane. This drawing on the top plane is 
called the " plan." On the front of the cube the " front view " or 
" front elevation " is drawn, and on the right side of the cube is 



ISTatuee of Oktiiograpiiic Projection 5 

the " right side elevation/' Three other views are supposed to be 
drawn on the other faces of the cube, but they are shown on Fig. 
2a, which is the perspective view of the cube from the opposite 
point of view, that is, from the back and from below instead of 
from in front and from above. 

This method of putting the object to be drawn in the center of 
a cube of transparent planes of projection is a device for the im- 
agination only. It explains the nature of the " projections," or 
" views," which are used in engineering drawing. 

5. Development or Flattening Out of the Planes of Projection. — 
Xow imagine the six sides of the cube to be flattened out into one 
plane forming a grouping of six squares as in Fig. 3. What we 



H 



IJ 

Plan 



§' 



X Y ® 



S 



IN 



Left Side 
Elevat/on 



^ Front 
Elevation 



(Right) Side 
Elevation 



V 



■A 



Back 
Elevation 



M' 



n 



Fig. 3. 



BOTTOM 

View 



have now is a descriptive or mechanical drawing of the object 
showing six " views." The object itself is now dispensed with and 
its projections are used to represent it. These six views are what 
we call the " regular views." With one slight change they cor- 
respond to the regular set of drawings of a house which architects 
make. 



Engineering Descriptive Geometry 



The set of six " regular " projections would not be altered by 
passing the transparent planes at unequal distances from each 
other, so long as they surround the object and are mutually per- 
pendicular. They may form a rectangular parallelopiped instead 
of a cube without altering the nature of the views. 

It will be noticed also that views on opposite faces of the cube 
differ but little. Corresponding lines in the interior may in one 
case be full lines and in the other "broken lines." Broken lines 
(formed by dashes about |" long, with spaces of tV") represent 
parts concealed by nearer portions of the object itself. All edges 
project upon the plane faces of the cube, forming lines on the draw- 





M 


Y, 














- 
F 


LAN 




X 


V 







S 


X 














<l 




El 


Front 
EVAT/ 


ON 
Z 


(RiG 
ElE 


HT) 5l 
VATiC 


N 





H 


Y 




S 














^ ^ 






L 


Plan 


(Right) Side 
Elevation 2- 


X 


V 













- 






El 


Front 
EVATl 


ON 





Fig. 4. 



Fig. 5. 



ings, the edges concealed by nearer portions of the object forming 
broken lines. 

6. The Reference Planes and Principal Views. — In drawings of 
parts of machinery six regular views are usually unnecessary. The 
three views shown in Pig. 3 are the " Principal Views," and others 
are needed only occasionally. The planes of those views are the 
" Eeference Planes." 

These views, when flattened from their supposed position about 
the object into one plane, give the grouping in Fig. 4. 

Another arrangement of the same views, obtained by unfolding 
the planes of the cube in a different order, is shown in Fig. 5. 
These two arrangements are standard in mechanical drawing, and 
are those most used. 



IsTatuke of Okthographio Projection 7 

7. The Nomenclature. — The nomenclature adopted is as follows : 
The " Eeference Planes/' or three principal planes of projection, 
are called from their position, the Horizontal Plane, or Ifi, the 
Vertical Plane, or V? and the (right) Side Plane, or g. The plane 
S is by some called the "Profile Plane." The point (Fig. 2), 
in which they meet, is the " Origin " of coordinates. The line 
OX, in which H and V intersect, is called the " Axis of X," or 
" Ground Line." The line OY, in which H and g meet, is called 
the " Axis of Y," and the line OZ, in which V and S meet, is 
called the " Axis of ZJ" The three axes together are called the 
" x4k.xes of Projection." 

Since drawings are considered as held vertically before the face, 
it is considered that the plane V coincides at all times with the 
" Plane of the Paper." In unfolding the planes from their posi- 
tions in Fig. 2 to that in Fig. 4, it is considered that the plane W 
has been revolved about the axis of X (line OX), through an angle 
of 90°, until it stands vertically above V- In the same way S is 
considered to be revolved about the line OZ, or axis of Z, until it 
takes its place to the right of V- 

The arrangement in Fig. 5 corresponds to a different manner of 
revolving the plane S- It is revolved about the axis of Y (OF) 
until it coincides with the plane f\, and is then revolved with H? 
about the axis of X, until both together come into the plane of the 
paper, or V- 

The three other faces of the original cube of planes of projection 
are appropriately called Jil', \', and S'- On account of the simi- 
larity of the views on them, to those on H, V and S? they are but 
little used. S' alone is fairly common since a grouping of planes 
W, V and S' is at times more convenient than the standard group 
H, V and S. 

8. Meaning of " Descriptive Geometry." — The aim of Engineer- 
ing or Mechanical Drawing is to represent the shapes of solid 
objects which form parts of structures or machines. It shows 
rather the shapes of the surfaces of the objects, surfaces which are 
usually composed of plane, cylindrical, conical, and other surfaces. 
In the drawing room, by the application of mathematical laws and 
principles, views are constructed. These are usually Plan, Front 



8 Engineering Descriptive Geometry 

Elevation, and Side Elevation, and are exactly such, views as would 
be obtained if the object, itself were put within a cage of trans- 
parent planes, and the true projections formed. 

It is these mathematical laws or rules which form the subject 
known as Descriptive Geometry. A drawing made in such a way 
as to bring out clearly these fundamental laws of projection, by the 
use of axes of projection, etc., may be conveniently called a " De- 
scriptive Drawing.'' 

In the practical application of drawing to industrial needs, 
short-cuts, abbreviations, and special devices are much used (their 
nature depending on the special branch of industry for which the 
drawing is made). In addition, the axes of projection are usually 
omitted or left to the imagination, no particular effort being made 
to show the exact mathematical basis provided the drawing itself 
is correct. Such a drawing is a typical " Mechanical Drawing." 
By the addition of axes of projection and similar devices, it may 
be converted into a strict " Descriptive Drawing." 

9. The Descriptive Drawing of a Point in Space. — The imagi- 
nary process of making a descriptive drawing consists in putting 
the object within a cube of transparent planes, and projecting 
points and lines to these planes. In practice the projections are 
formed all on a single sheet of paper, which is kept in a perfectly 
flat shape, by the application of rules of a geometrical kind de- 
rived from the imaginary process. The key to the practical pro- 
cess is in these rules. The first subject of exact investigation 
should be the manner of representing a point in space by its pro- 
jections and the fixing of its position as regards the " reference 
planes " by the use of coordinate distances. 

Eigs. 6 and 7 show the imaginary and the practical processes of 
representing P by its projections, 

Eig. 6 is a perspective drawing showing the cube of planes, or 
rather the three sides of the cube regularly used for reference 
planes. The cube must be of such size that the point P falls well 
within it. The perpendicular projectors of P are PPn, PPv and 
PPg. The origin and the axes of projection are all marked as on 
Eig. 2. 



Nature of Orthographic Projection 



In Fig. 7 the " field ^' of the drawing, that part of the paper 
devoted to it, is prepared by drawing two straight lines at right 
angles to represent the axes of projection, lettering the horizontal 
line XOYs and the vertical one ZOYh. This field corresponds to 
that of Fig. 4, the outer edges of the squares being eliminated 
since there is no need to confine each plane to the size of any par- 
ticular cube. If more field is needed, the lines are simply ex- 
tended. It must be remembered that these axes are quite different 
from the coordinate axes used in plane analytical geometry, or 
graphic algebra. These divide the field of the drawing into four 



^ 




In:: 



H 



X fe ^ 



Pv * 9 
V z 



/\ 



\ 



/ 



TTf. 



4 

Pc 



Fig. 6. 



Fig. 7. 



quadrants, of which three represent three different planes, mutu- 
ally perpendicular, the fourth being useful only for the purposes 
of construction. 

Usually the upper left quadrant, the " ISTorth-West," represents 
H ; the lower left quadrant, or " South-West," represents Y, and 
the lower right quadrant, or " South-East," represents S- 

On occasion the axes may be lettered XOZg horizontally and 
ZvOY vertically, to correspond to Fig. 5, the upper right quadrant 
now representing S. 

10. Coordinates of a Point in Space. — A point in space is 
located by its perpendicular distances from the three planes of 
projection, that is to say, by the length of its projectors. These 



10 Engineering Descriptive Geo]\ietrt 

distances are called the coordinates of the point, and are designated 
by X, y and z. In the example given, these values are 2, 3 and 1. 
In Fig. 6 PPs, the S projector of P, is. two units long, or x = 2. 
The perpendicular distance to the plane \, the V projector, PPv, 
is three units long. i/=3. In the same way PPfi, the H projector, 
is one unit long. 2 = 1. 

In describing the point P, it is sufficient to state that it is the 
point for which x=2, y = o, and z = l. This is abbreviated con- 
veniently by calling it the point P (2, 3, 1), the coordinates, given 
in the bracket, being taken always in the order x, y, z. 

The projectors, the true coordinate distances, are shown in Fig. 
6 by lines of dots, not dashes. 

If in each plane W, \ and S? perpendicular lines are drawn 
(dashes, not dots) from the projections of P to the axes, we shall 
have the lines Pne and P;,/, P^e and P^g, Psg and Pgf. These lines 
meet in pairs at e, g, and f, forming a complete rectangular paral- 
lelepiped of which P and are the extremities of a diagonal. The 
other corners of the parallelopiped are P?,, Pv, Ps, &, f and g. 

Each coordinate, x, y and z, appears in four places along four 
edges of the parallelopiped, as is marked in Fig. 6. 

The distances x, y and z are all considered positive in the case 
shown. 

In Fig. 7, the descriptive drawing of the point P, P itself does 
not appear, being represented by its projections, Pn, Pv and Pg. 
The true projectors (shown in Fig. 6 by lines of dots) do not 
appear, but in place of each coordinate three distances equal to it 
do appear, so that in Fig. ^ x, y and z each appear in three places 
as is there marked. Thus x appears as Pixfn, eO, and Pvg. As all 
these are measured to the left from the vertical axis, ZOYn, it 
follows that P],epv is a straight line, or Pn is vertically above P^. 
It is often said that P« " projects vertically " to Ph. In the same 
way Pu " projects horizontally " to Pg. The distance y appears as 
ePj,, Ofh, Ofs, and gPs. The point / appears double due to the 
axis of Y itself doubling. To represent the original coincidence of 
fh and fs, a quadrant of a circle with center at is often used to 
connect them. 



Nature of Orthogkaphic Projection 11 

11. Three laws of Projection for ff, V and 5.— The three rela- 
tions shown by Fig. 7 amonnt to three laws governing the pro- 
jections of a point in the three views, and must always be rigidly 
observed. They may seem easy and obvious when applied to one 
point, but when dealing with a multitude of points it is not easy 
to observe- them unfailingly. 

They may be thus tabulated : 

(1) Ph must be vertically above Pv. 

(2) Ps must be on the same horizontal line as Pv. 

(3) Ps must be as far to the right of OZ as Ph is above OX. 
From these laws it follows that if two projections of a point are 

given, the third is easily found. In Fig. 7, if two of the corners 
of the figure Pj,fjifsPsPv are given, the figure can be graphically 
completed. Much of the work of actual mechanical drawing con- 
sists in correctly locating two of the projections of a point by plot- 
ting or measuring, and of finding the other projection by the appli- 
cation of these laws or of this construction. Constant checking 
of the points between the various views of a drawing is a vital prin- 
ciple in drawing. 

On the drawing board the horizontal projection of Pv to Ps is 
naturally done by the T-square alone, and the vertical projection 
of Pft to Pv by T-square and triangle. ' There are two methods of 
carrying out the third law in addition to the graphical construc- 
tion of Fig. 7. Fig. 8 shows a gi^aphical method which makes use' 
of a 45° line, OL, in the construction space, instead of the quad- 
rant of a circle. It is easier to execute, but the meaning is not so 
clearly shown. The third method is by the use of the dividers, 
directly to transfer the x coordinate from whichever place it is. 
fi'rst plotted, to the other view in which it appears. 

12. Paper Box Diagrams. — When studying a descriptive draw- 
ing, such as Fig. 8, imagine as you look at Pv that the real point P 
lies haclv of the paper, at a distance equal to ePn. 

Whenever figures in the text following seem hard to grasp, carry 
out the following scheme. Trace the figure on thin paper, or on 
tracing cloth. Using Fig. 8 as an example, and supposing it to 
have been traced on semitransparent paper, hold the paper before 
you and fold the top half back 90° on the line XOYs. Then, view- 



12 



Engineering Descriptive Geometry 



ing Pft from above, imagine the true point P to lie below the paper 
at a distance equal to ePv, in the same way as you imagine P to 
lie back of Pv at a distance equal to ePn. 

After flattening the paper, fold the right half back 90° on the 
line ZOYh, and, viewing Pg from the right, imagine P to lie back 
of Ps a distance Pvg. Finally, crease the paper on the line OL, 
OL itself forming a groove, not a ridge, and bend the paper on all 




Fig. 8. 



the creases at once, so that ff and S fold back into positions at 
right angles to V aiid to each other at the same time. 

The "construction space" YhOYs is thus folded away inside 
and OYh and OYs come in contact with each other. Fig. 9 shows 
the final folding partly completed. 

No diagram, however complicated, can remain obscure if studied 
from all sides in this manner. 

To have a convenient name, these space diagrams may be called 
" Paper Box Diagrams.'' 



Natdee of Orthographic Projection 



13 



Figs. 4 and 5 make good paper box diagrams, while Fig. 3 may 
be traced and folded into a perfect cube which, if held in proper 
position, will give the exact views shown in Figs. 2 and 2a, omit- 
ting the solid object supposed to be seen in the center of those 
figures. 

13. Zero Coordinates. — Points having zero coordinates are some- 
times perplexing. If one coordinate is zero, the point in question 
is on one of the reference planes, and indeed coincides with one of 
its own projections. Since x is the length of the orthographic 
projector of the point P upon the plane S? if x=(), this projector 
disappears and the point P and its S projection Ps coincide. If 




Fig. 9. 



the point Q (0, 3, 1) is to be plotted it will be found to coincide 
with Pg in Pig. 6, The descriptive drawing will correspond with 
Fig. 7 with all lines to the left of ZOYn omitted, and with the let- 
tering changed as follows: For Pg put Qs (and Q) , for //; put Qn, 
for g put Qv. The student should make this diagram on cross- 
section paper and should study out for himself the similar cases for 
the points Q' (2, 0, 1) [P„ in Fig. 6] and Q" (2, 3, 0) [P^ in Fig. 
6] and should proceed from them to more general cases, assuming 
ordinates at will, using cross-section paper for rapid sketch work 
of this kind. 

If two coordinates are zero, the point lies on one of the axes, 
on that axis, in fact, which corresponds to the ordinate which is not 
zero. Thus the point E (2, 0, 0) is the point e of Fig. 6, En and 
Ev are at e, and Eg is at 0. 




Fig. 10. 




Fig. 10a. 

These wire-mesh cages are not essential for a clear understanding 
of the course. Cross-section paper should be used in the solution 
of the problems and folded to make " space " or " paper box " dia- 
grams, to illustrate knotty points. These folded diagrams are 
practically miniature cages. The full-size cages are very con- 
venient for class-room demonstrations. 



16 Engineering Descriptive Geometry 

Wire-mesh Cage. 

If possible^ it is very desirable to have cages similar to Fig. 10, 
formed of wire-mesh screens, representing the planes ff, Y, S and 
S'. On these screens chalk marks may be made and the planes, 
being hinged together, may afterward be brought into coincidence 
with V^ as represented in Fig. 10a. 

In order to plot points in space within the cage, pieces of wire 
about 20 inches long, with heads formed in the shape of small 
loops or eyes, are used as point marJcers. They may be set in holes 
drilled in the base of the cage at even spaces of 1" in each direc- 
tion, so that a marker may be set to represent any point whose x 
or y coordinates are even inches. To adjust the marker to a re- 
quired z coordinate, it may be pulled down so that the wire projects 
through the base, lowering the head the required amount, z may 
vary fractionally. 

In Fig. 10 a point marker is set to the point P (11, 4, 6), and 
the lines on the screens have been put on with chalk, to represent 
all the lines analogous to those of Fig. 6. 

Fig. 10a represents the descriptive drawing produced by the 
development of the screens in Fig. 10. It is analogous to Fig. 7. 

Several points may be thus marked in space and soft lead wire 
threaded through the loops, so that any plane figure may be shown 
in space, and its corresponding orthographic projections may be 
drawn on the planes in chalk. 

Problems I, 

(For solution with wire-mesh cage.) 

1. Plot by the use of the wire markers the three points. A, B 
and C, whose coordinates are (5,12,11), (3,3,3), and (12,4,8), 
and draw the projections on the screens in chalk. By joining point 
to point a triangle and its projections are formed. Use lead wire 
for joining the points, and chalk lines for joining the projections. 

2. Form the triangle as above with the following coordinates: 

(11,3,2), (12,6,12) and (14,12,7). 

3. Form the triangle as above with the following coordinates : 

(7,0,11), (9, 9,0) and (2,2,3). 



Nature of Orthogeaphic Projection 17 

4. Form the triangle as above with the following coordinates: 

(0,11,13), (14,3,3) and (14,13,0). 
(The following examples may be solved on coordinate paper, or 
plotted in inches on the blackboard.) 

5. Make the descriptive drawing of a triangle in three views by 
plotting the vertices and joining them by straight lines. The 
vertices are the points A (1, 10, 8), B (5, 6, 8), C (9, 2, 4). 

6. Make the descriptive drawing as above using the points 

A (12, 2, 5), 5 (0, 8, 6), (7 (4, 6, 0). 

7. Make the descriptive drawing as above using the points 

A (3,4,2), B (13,8,10), C (5,10,14). 

8. The four points A (3, 3, 3), B (3, 3, 15), C (15, 3, 15), and 
D (15, 3, 3) form a square. Make the descriptive drawing. Why 
are two projections straight lines only? What are the coordinates 
of the center of the square? 

9. The four points A (12,2,12), B (2,2,12), C (7,14,12), 
and D (7, 6, 2) are the comers of a solid tetrahedron. Make the 
descriptive drawing, being careful to mark concealed edges by 
broken lines. 

10. Make the descriptive drawing of the tetrahedron A (2, 3, 2), 
B (9, 8, 3), C (4, 8, 9), Z> (12, 3, 6), marking concealed edges by 
broken lines. 

11. Make the descriptive drawing of the tetrahedron A (3, 2, 4), 
B (6,8,2), C (8,1,8), D (2,7,8). 

12. Plot the points A (12,7,7), B (8,13,5), C (2,9,2), and 
D (6,3,4). Why is the V projection a straight line? 

13. Make the descriptive drawing of the tetrahedron A (13, 5, 3), 
B (1, 5, 3), C (7, 2, 6), D (7, 8, 6). To which axis is the line AB 
parallel? To which axis is CD parallel? 

14. Plot and join the points A (11, 3,3), B (3, 3, 3), C (7, 9, 7), 
and Z)(15, 9, 7). Bo AC and BD meet at a point or do they pass , 
without meeting? 



CHAPTEE 11. 



ORTHOGRAPHIC PROJECTION OF THE FINITE STRAIGHT 

LINE. 

14. The Finite Straight Line in Space: One not Parallel to any 
Reference Plane, or an " Oblique Line." — A line of any kind con- 
sists merely of a succession of points. Its orthogi'aphic projection 
is the line formed by the projections of these points. 

In the case of a straight line, the orthographic projection is 
itself a straight line, though in some cases this straight line may 
degenerate to a single point, as mathematicians express it. 





■¥— i— -L-S- 




I I I 'Y, 




Fig. 11. 

To find the ]HI, V and S projections of a finite straight line in 
space, the natural course is to project the extremities of the line 
on each reference plane and to connect the projections of the ex- 
tremities by straight lines. We shall not consider this as requir- 
ing proof here. It is common knowledge that a straight line cannot 
be held in any position that will make it appear curved, and ortho- 
graphic projection is, as shown by Fig. 1, only a special case of 
perspective projection. The strict mathematical proof is not ex- 
actly a part of this subject. 



Okthogeaphic Projection of Finite Straight Line 19 



The projectors from the different points of a straight line form 
a plane perpendicular to the plane of projection. This " projector- 
plane/" of course, contains the given line. If the straight line is 
a limited or finite line the projector-plane is in the form of a 
quadrilateral having two right angles. Thus in Fig. 11 the ff 
projectors of the straight line AB form the figure AAj,BnB, having 
right angles at An and Bji. These projector-planes AA]iBi,B, 
AAsBsB, and AAvBi-B are shown clearly in this perspective draw- 
ing, in which they are shaded. 

Fig. 12 is the descriptive drawing of the same line AB which has 
been selected as a " line in space," that is, as one which does not 
obev any special condition. In such general cases the projections 
are all shorter than the line itself. As drawn, the extremities are 
A (1,1,5) and B (5,1,2; 






% 






B>\ 


• 






IHI ^ 

1 1 1 


1 I 






X ' ' ' 


' ' o 


' ■ ■ ■ ■ y. 


V 


A, 
z. 


As 


B. 



Fig. 13. 



Fig. 14. 



15. Line Parallel to One Reference Plane, or Inclined Line. — 

A line which is parallel to one reference plane, but is not parallel 
to an axis, appears projected at its true length on that reference 
plane only. 

Figs. 13 and 14 show a line five units long, connecting the points 
A (1,2,2) and B (5,5,2). Ai>Bk is also five units in length but 
ArB,- is but four and AsBg is three. The projector-plane AAnBhB 
is a rectangle. 

The student should construct on coordinate paper the two simi- 
lar cases. For example : the line C {4,2,1), D (1, 2, 5) is parallel 
to V; ^ (2, 1,2), F (2, 5, 5) is parallel to S- 



20 



Engineeeing Descriptive Geometry 



16. Line Parallel to One of the Axes and thus Parallel to 
Two Reference Planes. — If a finite straight line is parallel to one 
of the axes of projection, its projection on the two reference planes 
which intersect at that axis, will be equal in length to the line 
itself. Its projection on the other reference plane will be a single 
point. 

Fig. 15 is the perspective drawing and Fig. 16 the descriptive 
drawing, of a line parallel to the axis of X, four units in length. 
Its extremities are the points A (1,2,3) and 5 (5, 2, 3). In H 




M 


\ 




^K 


K. 


\ 


x' ' ' ' 


' ' 


_ ' ' ' 'Y. 


Bv 


Av 


A.&Bs 


V 


z' 


B> 



Fig. 15. 



Fig. 16. 



and V ifs projections are four units long. The projector-planes 
AA]iB],B and AAvBvB are rectangles. The S projector-plane de- 
generates to a single line BAAg. It will be seen that the coordi- 
nates of the extreme points of the line differ only in the value of 
the X coordinate. In fact, any point on the line will have the 
y and z coordinates unchanged. It is the line {x variable, 2, 2). 

The student should construct for himself descriptive drawings 
of lines parallel to the axis of Y and the axis of Z, using prefer- 
ably " coordinate paper " for ease of execution. Good examples 
are the lines C (1,1,1), D (1,5,1) and ^ (3, 1, l),i^ (3,1,4). 
Points on the line OD differ only as regards the y coordinate. It 
is a line parallel to the axis of Y. EF is parallel to the axis of Z 
and z alone varies for different points along' the line. 



Orthographic Projection of Finite Straight Line 21 

17. Foreshortening. — The projection of a line oblique to the 
plane of projection is shorter than the original line. This is 
called foreshortening. The ff, V and S projections of Fig. 12, 
and the V and S projections of Fig. 14, are foreshortened. It is 
a loose method of expression, but a common one, to say that a line 
is foreshortened when it is meant that a certain projection of a 
line is shorter than the line itself. "When subscripts are omitted 
and AhBh is called AB, it is natural to speak of the line AB as 
appearing "foreshortened" in the plan view or projection on ff. 
This inexact method of expression is so customary that it can 
hardly be avoided, but with this explanation no misconception 
should be possible. 

18. Inclined and Oblique Lines. — The words Inclined and Obli- 
que are taken generally to mean the same thing, but in this subject 
it becomes necessary to draw a distinction, in order to be able to 
specify without chance of misunderstanding the exact nature of a 
given line or plane. 

A line will be described as : 
Parallel to an axis, when parallel to any axis. As a special case a 

line parallel to the axis of Z may be called simply vertical. 
Inclined, when parallel to a reference plane, but not parallel to an 

axis. The line AB, Fig. 13, is an illustration. 
Oblique, when not parallel to any reference plane or axis. The 

typical " line in space " is oblique. AB, of Fig. 11, illustrates 

this case. 

19. Inclined and Oblique Planes. — A plane will be called: 
Horizontal, when parallel to H- The V projector-plane in Fig. 15 

is of this kind. 
Vertical, when parallel to V or S- The ff projector-plane in Fig. 

15- is of this kind. 
Inclined, when perpendicular to one reference plane only. The H 

projector-plane of Fig. 13 is of this kind. 
Oblique, when not perpendicular to any reference plane. Planes 

of this kind will appear later on. 
The surface of the solid of Fig. 2 is composed of vertical, hori- 
zontal, and inclined planes (but no oblique plane). Its edges are 
3 



22 



Engineerixg Descriptive Geometry 



lines, parallel to the axes of X, Y and Z ; and inclined lines (be- 
cause parallel to S) ^ but no oblique lines. 

20. The Point on a Given Line. — It is self-evident that if a 
given point is on a given line, all the projections of the point must 
lie on the projections of the line. 

If the middle point of a line AB is projected, as C in Fig. 17, its 
projections Cji, Cv, and Cs bisect the projections of the line. The 
reason for this appears when we consider the true shape of the 




Fig. it 



projector-planes, all three of which appear distorted in the per- 
spective drawing. Fig. 17, and which do not appear at all on the 
descriptive drawing, Fig. 18. In Fig. 17 AAnBnB is a quadri- 
lateral, having right angles ai An and Bk, it is therefore a trape- 
zoid. CCn is parallel to AAn and BBn, and since it bisects AB at C, 
it must also bisect A^Bh at Cn. The result of this is that in Fig. 
18, where the projections which do appear are of their true size,. 
Cn bisects AnBii, Co bisects AvBv, and Cs bisects AgBg. 

This principle applies to other points than the bisector. Since 
all W projectors are parallel to each other, if any point divides AB 
into unequal parts, the projections of the point will divide the 
projectors of AB in parts having the same ratio. A point one- 



Oethogeaphic Projection of Finite Straight Line 23 

tenth of the distance from A to B will, by its projections, mark 
off one-tenth of the distance on AnBh, AvBv, etc. 

The points illustrated in Figs. 17 and 18 are A (3,3,1), 
B (5, 5, 5) and C (3|, 4, 3). It will be noticed that the x coordi- 
nate of C is the mean of those of A and B, and the y and z coordi- 
nates of C also are the mean of the y and z coordinates of A and B. 

Unless all three of the projections of a point fall on the pro- 
jections of a line, the point is not in the given line. If one of the 
projections of the point be on the corresponding projection of the 
line, one other projection of both point and line should be ex- 
amined. If in this second projection it is found that the point 
does not lie on the line, it shows that the point in space lies in one 
of the projector-planes. 

Thus the point D in Fig. 18 has its V projection on AvB^, but 
its ff and S projections are not on AnBh and AgBg. D is not a 
point in the line AB but is on the V projector-plane of AB, as is 
clearly shown on Fig. 17. 

In the case illustrated, Dv bisects BvCv. The plotting of the V 
projection of a point is governed only by its x and z coordinates. 
Dv bisects BvC'v because its a; and 2 coordinates are the means of 
the X and z coordinates of B and C. The y coordinate of D, how- 
ever, has no connection with the y coordinates of B and C. 

21. The Isometric Diagram. — A device to obtain some of the 
realistic appearance of a true perspective drawing without the 
excessive labor of its construction is known as " isometric " draw- 
ing, 

A full explanation of this kind of drawing will follow later, 
but for present purposes we may regard it as a simplified per- 
spective of a cube in about the position of that in Figs. 2, 6, 11, 
etc., but turned a little more to the left. Vertical lines are un- 
changed. Lines which are parallel to the axis of X, and which in 
the perspective drawing incline up to the left at various angles, are 
all made parallel and incline at 30° to the horizontal. In the 
same way lines parallel to the axis of Y are drawn at 30° to the 
horizontal, inclining up to the right. 



24 



Engineering Descriptive Geometry 



Fig. 19 shows the shape of a large cube divided into small imit 
cubes. In plotting points the same scale is used in all three direc- 
tions, that is, for distances parallel to all three axes. Fig. 19a 
shows the point P (2, 3, 1) plotted in this manner, so that the 
figure is equivalent to the true perspective drawing. Fig. 6. 

It is not intended that the student should make any true per- 
spective drawing while studying or reciting from this book. If any 
of the space diagrams here shown by true perspective drawings 




Fig. 19. 



Fig. 19a. 



must be reproduced, the corresponding isometric drawing should 
be substituted. 

For rapid sketch work, especially ruled paper, called " isometric 
paper,'^ is very convenient. It has lines parallel to each of the 
three axes. With such paper it is easy to pick out lines correspond- 
ing to those of Fig. 19. 

An excellent exercise of this kind is to sketch on isometric 
paper the shaded solid shown in Fig. 2, taking the unit square of 
the paper for 1" and considering the solid to be cut from a 10" cube, 
the thickness of the walls left being 2", and the height of the tri- 
angular portion being 6". The solid may be sketched in several 
positions. 



Okthogeaphic Projection of Finite Straight Line 25 

Problems II. 

(For solution with wire-mesh cage, or cross-section paper, or on 

blackboard.) 

15. A line connects the points A (5,2,6) and B (5, 12, G). 
What are the coordinates of the point C, the center of the line? 
What are the coordinates of D, a point on the line, one-tenth of 
the way from A to B? 

16. Same with points A (6, 6, 2) and B (6, 6, 12). 

17. Draw the line AB whose extremities are A (2, 7, 4) and 
B (14, 2, 4). On what view does its true length appear? What is 
this length? What are the coordinates of a point G on the line 
one-third of its length from A ? 

18. With the same line A (2,7,4), B (14,2,4), state what is 
the true shape of the H projector-plane. Give length of each edge 
and state what angles are right angles. Same for V projector- 
plane. 

19. Same as Problem 18, with line A (4, 2, 2), 5 (4, 11, 8). 

20. With the line of Problem 19, state what is the true shape of 
the ff and S projector-planes, giving length of each edge, and 
state which angles are right angles. 

21. Same as Problem 17, with line A (0, 4, 8), B (9, 4, 1). 

22. The H projection of C (8, 2, 6) lies on the JHI projection of 
the line A (10, 1, 9), B (2, 5, 2). Is the point on the line? 

23. Same as Problem 22, with line A (2, 1,8), B (8, 10, 5), and 
point C (4,4,7). 

24. A triangle is formed by joining the points A (6, 3, 1), 
B (10,3,10) and C (2,3,10). In what view or views does the 
true length of AB appear? In what view or views does the true 
length of BC appear? Mark the center of the triangle (one-third 
the distance from the center of the base BC to the vertex A) and 
give its coordinates. 

25. Same with points A (5,9,6), B (5,3,1) and C (5,3,11). 

26. Same with points A (10, 1, 4), B (7, 10, 4) and C (1, 4, 4). 

27. The V projections of the points A (8, 1, 2), S (10, 3, 8), 
C (4, 3, 10) and D (2, 1, 4) form a square. Draw the projections 
and connect them point to point. What are the coordinates of the 
center where AC and BD intersect? 



26 Engineering Desckiptive Geometky 

28. Plot the parallelogram A (11,3,3), B (3,3,3), C (7, 9, 7), 
D (15, 9, 7). The diagonals intersect at E. Give the coordinates 
of E. Describe the ff projector-planes of AB, AC, and CD, giving 
the length of each end projector. Is the plane of the figure in- 
clined or oblique? Is AC an inclined or an oblique line? 

29. Plot the quadrilateral 

A (11,10,3), B (3,10,11), (7,2,7), D (11,4,3). 

Is the plane of the figure horizontal, vertical, inclined or oblique? 

Is the line AB horizontal, vertical, inclined or oblique? 

Is the line BG horizontal, vertical, inclined or oblique? 

Is the line CD horizontal, vertical, inclined or oblique? 

Is the line DA horizontal, vertical, inclined or oblique? 



CHAPTEE III. 
THE TRITE LENGTH OF A LINE IN SPACE. 

22. The Use of an Auxiliary Plane of Projection.— To find the 
true length of a " line in space/' or oblique straight line, an auxil- 
iary plane of projection is of great value, and is constantly used 
in all branches of Engineering Drawing. 

A typical solution is shown by Figs. 20 and 21. The essential 
feature is the selection of a new plane of projection, called an 





Fig. 20. 



Fig. 21. 



auxiliary plane, and denoted by U, which must be parallel to the 
given line and easily revolved into coincidence with one of the 
regular planes of reference. 

This auxiliary plane is passed parallel to one of the projector- 
planes. In Fig. 20 the plane S' of the cube of planes has been 
replaced by a plane U, parallel to the H projector-plane, AAnBnB. 
Like that plane, U is also perpendicular to H, and XM, the line 
of intersection of U and H, is parallel to 4,5,, The distance of 
the plane U from the projector-plane may be taken at will and 
in the practical work of drawing it is a matter of convenience, 
choice being governed by the desire to make the resulting figures 
clear and separated from each other. In Fig. 20 the auxiliary 



28 Engineering Descriptive Geometry 

plane \] has been established by selecting a point X in H for it 
to pass through. \] is an " inclined plane/' not an " oblique plane." 

23. Traces of the Auxiliary Plane U. — ^The auxiliary piano U 
cuts the plane V in a line XN, parallel to the axis of Z. The 
lines of intersection of \] with the reference planes^, are called the 
" traces " of \]. Since there are three reference planes, there may 
be as many as three traces of \]. In the case illustrated in Fig. 20, 
there are, however, but two traces. Only one of these three possible 
traces of \] can be an inclined line. In Fig. 20 the trace XM alone 
is an "inclined" line. 

We shall see later that the auxiliary plane may be taken per- 
pendicular to V or to S as alternative methods. In every case 
there is but one inclined trace, that on the plane to which UJ is 
perpendicular. It is this trace which has the greatest importance in 
the process. For the sake of uniformity, M and N will be assigned 
as the symbols for marking the traces of an auxiliary plane of 
projection. 

24. The U Projectors. — A new system of projectors, AAu, BBu, 
etc., project the line AB upon the plane U- These projectors, 
being perpendicular distances between a line and a plane parallel to 
it, are all equal, and the projector-plane AAuBuB of Fig. 20 is in 
reality a rectangle. AuBu is therefore equal in length to AB, or 
AB is projected upon \J, without foreshortening. 

25. Development of the Auxiliary Plane U- — The descriptive 
drawing. Fig. 21, is the drawing of practical importance, which is 
based on the perspective diagram. Fig. 20, which shows the mental 
conception of the process employed. In practical work, of course. 
Fig. 21 alone is drawn, and it is constructed by geometrical reason- 
ing deduced from the mental process exhibited by Fig. 20. 

In the process of flattening out or " developing " the planes of 
projection, U is generally considered as attached or hinged to the 
" inclined trace," XM in this example. In Fig. ' 21 \] has been 
revolved about XM, bringing it into the plane of H, the trace XN 
having opened out to two lines. N separates into two points and is 
marked Nu as a point in U and Nv as a point in V, analogous to 
Y}i and Yg in the development of the reference planes. The space 
NuXNv, like Yi,OYs, may be considered as construction space. 



The True Length of a Line in Space 29 

26. Fourth Law of Projection — that for Auxiliary Plane, \]. 

It will be seen from Fig. 20 that AA^eA^ is a rectangle and that 
eAv is equal to AiA. On the descriptive drawing, Fig. 21, these 
two lines, eAv and eA],, form one line perpendicular to OX. This 
is in accordance with the first law of projection of Art. 11. 

As the plane U is perpendicular to ff we have the same rela- 
tion there, and AAuhAu, Fig. 20, is a rectangle. M« is therefore 
equal to AuA, and in the development. Fig. 21, Auh and TcAn form 
one line AukAn perpendicular to XM. 

If from Au and Av, Fig. 20, perpendiculars are let fall upon the 
intersection of U and V (the trace XN) they will meet at the 
common point I, both Ti:AulX and XlAvS being rectangles. In the 
descriptive drawing. Fig. 21, AJ is perpendicular to XNu, hh is 
the arc of a circle, center at X, and l^Av is perpendicular to XNv. 
The following law of projection governs the position of Au in 
the plane U* 

(4) From the regular projections of A draw perpendiculars 
to the traces of U. These lines continued . into the field 
of U intersect at Au- One of these lines is carried across 
the construction space by the arc of a circle whose center 
is the meeting point of the traces of U- 

27. The Graphical Application of this Law to a Point. — The 
procedure for locating the projection Au on the descriptive drawing. 
Fig. 21, after the location of the plane \] has been determined, is 
as follows : From the adjacent projections of the point draw lines 
perpendicular to the traces of the plane U- Continue one of these 
lines across the trace. Swing the foot of the other perpendicular 
to the duplicated trace, and continue it by a line perpendicular to 
this trace to meet the line first mentioned. Their intersection is 
the projection of the point on \]. In Fig. 21, this requires AjJcAu 
to be drawn perpendicular to XM, and the line AvlvhAu to be 
traced as shown. 

28. The True Length of a Line. — The procedure for finding the 
true length of a line consists in first drawing, Fig. 21, a line paral- 
lel to one of the projections of the line to act as the trace of the 
auxiliary plane. Where this trace intersects an axis of projection 
perpendicular lines are erected, one perpendicular to the axis, one 



30 



Engineering Descriptive Geometry 



perpendicular to the trace. These lines are the two developed 
positions of the other trace of the plane \]. Then locate the ex- 
tremities of the given line on the auxiliary plane UJ. The line 
joining the extremities is the required projection of the line on \J, 
and is equal in length to the given line. 

29. Alternative Method of Developing the Auxiliary Plane, HJ. — 
A modification of this construction is shown in the descriptive 
drawing, Fig. 22, in which the plane U has been revolved about 
the vertical trace XN until it coincides with the plane V- ^^ 
separates into two lines, XMh and XM^. h, of Fig. 20, becomes 
Jck and hu, and the space MjtXMu is construction space. A is on 




Fig. 22. 



a horizontal line drawn through A^. Anhn is perpendicular to XMn. 
Ichku is the arc of a circle having X as a center, and kuAu is per- 
pendicular to XMu. Au is thus located. 

This method of development of the planes is much less common 
in practical drawing than the other, because, as a rule, it is less 
convenient than the first method. In such cases as occur it offers 
no particular difficulty. Both Figs. 21 and 22 are' solutions of 
the problem of finding the true length of the oblique line AB by 
projection on an auxiliary inclined plane, U- 

30. Alternative Positions of the Plane U-— We saw that the 
exact position of the plane \J, so long as it remained perpendicular 
to IHI and parallel to AnBn, was left to choice governed by practical 

4r 



The True Length of a Line in Space 31 

considerations. U itself, however, may be taken perpendicular to 
V and parallel to A^Bv, or it may be taken perpendicular to S and 
parallel to AsBa. To get an entire grasp of the subject the student 
is advised to trace Fig. 21 on thin paper, or plot it on coordinate 
paper, points A, B, and X being (6,6,2), (10,10,8) and 
(11, 0, 0), and fold the figure into a paper box diagram, the con- 
struction spaces NuXNv and YjiOYs' being creased in the middle 
and folded out of the way. Fig. 22 will serve equally well. The 
final result will be a paper box exactly similar to Fig. 20. 

The variation in which U is perpendicular to V ^ay be plotted, 
passing the new inclined trace of U (lettered YM) through the 
point (0, 0, 3) parallel to AvBv. Fold this figure into a paper box, 
the paper being cut along a line YN perpendicular to YM. 

The other variation may be plotted with the inclined trace of \] 
on the plane Sj parallel to AsBs and passing through the point 
(0,0,6) {fs of Fig. 21). Letter this trace YsM and draw YsN 
perpendicular to it, inclining up to the right. The paper must be 
cut on this line to enable it to be properly folded. 

31. The Method Applied to a Plane Figure. — The special value 
of this use of the auxiliary plane is seen when one operation serves 
to give the true length of a number of lines at once, and thus 
shows a whole plane figure in its true shape. 

In Fig. 23 the polygon ABODE is shown by its projection, the 
point A alone being lettered. It is noticeable that in V the edges 
all form one straight line. The V projector-planes of the various 
edges are therefore all parts of the same plane, and the polygon 
itself is a plane figure placed perpendicular to V- It may be said 
the polygon is " seen on edge " in V- 

An auxiliary plane U has been taken parallel to the plane of the 
polygon, and therefore perpendicular to V- The trace XM being 
parallel to the V projections of the edges, this auxiliary plane serves 
to show the true length of all the edges at once. The projection on 
U is the true shape of the polygon ABODE. In the case illustrated, 
the U projection discloses the fact that the polygon is a regular 
pentagon, a fact not realized from the regular projections, owing 
to the foreshortening to which they are subject. 



32 



Engineering Descriptive Geometry 



This figure is well adapted to tracing and folding into a paper 
box diagram. 




Fig. 23. 



32. The True Length of a Line by Revolving About a Projector. 

— A second method of finding the true length of a line seems in a 
way simpler, but proves to be of much less value in practical work. 
The method consists in supposing an oblique line AB to be revolved 
about a projector of some point in the line until it becomes parallel 
to one of the planes of reference. In this new position it is pro- 
jected to the reference plane as of its true length. 

In Fig. 24 the V projector-plane of the line AB has been shaded 
for emphasis (A is the point (1, 1, 5), and B is the point (5, 4, 2) ) . 
The projector AAv has been selected at will, and the V projector- 
plane (of which the line AB is one edge) has been rotated about 
AAv as an axis until it has come into the position AvB'vB'A. In 
its new position, AB' projects to ff as At,B'i,. This is the true 
length of the line. During its rotation the point B has moved to 
B', but in so doing it has not revolved about A as its center, but 
about the point h on AiA extended. hAv is equal in length to BBy. 



The True Lexgth of a Line ix Space 



33 



Bv moves to B'v, revolving about Av as a center. In Fig. 25, the 
corresponding descriptive drawing, the original projections are 
shown as full lines and the projections of the line after the rota- 
tion has occurred are shown by long dashes. 

In Vj -^vBv swings about Av as a pivot until in its new position 
AvB'v it is parallel to OX. In W, Bn moves in a line parallel to 
OX (since in Fig. 24 the motion of B takes place entxrely in the 
plane of hBB', which is parallel to V)? and as B'h must be verti- 
cally above B'v the motion terminates where a line drawn vertically 




Fig. 24 



up from B'v meets the horizontal line BhB'n. Joining Ah and B'n^ 
the new H projection is the true length of the given line. The S 
projection is of no interest in this case. The H and V projections 
of Fig. 25 show the graphical process corresponding to the theory 
of this rotation. In V, -^u moves' to B'v, whence a vertical pro- 
jector meeting a horizontal line of motion from Bh determines B'ji, 
the new position of Bh. AhB'h is the true length of the line. The 
arrow-heads on the broken lines make these steps clear. 

33. Variations in the Method. — The method is subject to wide 
variations. The same projector-plane AAvBvB, Fig. 24, revolving 
about the same projector AAv, might start in the opposite direction 
and swing to a position parallel to S- The graphical process of 
Fig. 25 would then confine itself to V and S instead of V and JrJ, 



34 



Engineeeing Desceiptive Geometet 



In addition, the rotation might have been about BBv as an axis 
or about the V projector of any point in AB or AB extended. 
Finally, the ff projector-plane or the S projector-plane might 
have been selected and made to revolve into position. There are 
six distinct varieties of the process, each one subject to great modifi- 
cations. 

This method can be applied to a plane figure which appears " on 
edge " in one of the regular views. In Fig. 26 a polygon lies in a 




Fig. 26. 



plane perpendicular to V- There are two varieties of the process 
applicable in this case. Choosing the V projector of the point A 
for the axis of rotation, the whole polygon may be rotated up par- 
allel to IHI, thence its true shape projected upon Jil', or it may be 
revolved down until parallel to S^ thence its true shape projected 
upon S- Both methods are shown, though of course in practice one 
at a time should be enough. 

34. A Projector-Plane ITsed as an Auxiliary Plane. — The two 
processes for finding the true length of a line differ in this respect. 



The True Length of a Line in Space 35 

In one the line is projected on a plane which is revolved into 
coincidefice with one of the reference planes, by revolving about a 
line in that reference plane. In the second process, a projection 
plane is itself revolved about a projector, that is, about a line 
perpendicular to one reference plane, to a position parallel to a 
second reference plane. The line in its new position is projected 
on the latter plane. 

A method which is a modification of the first process is in many 
cases very simple. A projector-plane is itself used as an auxiliary 
plane, and is revolved into coincidence with the plane to which it is 
perpendicular by rotation about its trace in that plane. In Fig. 23, 
for example, instead of passing XM parallel to AyCv, AvCv would be 
extended to the axis of X, and used itself for the inclined trace of 
the auxiliary plane. XN would be moved to the right and other 
slight modifications made. 

As in the second method, a projector-plane is here rotated; but 
it is not rotated about a projector, but about a projection (its 
trace), and the real similarity of the process is with the first 
method, that of the auxiliary plane of projection. It is but a 
special case of this kind. 

In practical drawing, it rarely happens that one of the projector- 
planes can be thus used itself with advantage as an auxiliary plane 
of projection. It leads usually to an overlapping of views and it 
will not be found so useful as the more general method. 

For the continuation of this study, all these methods should be 
at the students' finger ends. 

35. The True Length of a Line by Constructing a Right Tri- 
angle. — These methods of finding the true length of a line are 
generally used for the true lengths of many lines in one operation, 
or for the true shape of a plane figure. When a single line is 
wanted, the construction of a right triangle from lines whose true 
lengths appear on the drawing is sometimes resorted to. In Fig. 24 
the triangle ABh is a right triangle, AB being the hypotenuse and 
AhB the right angle. In the descriptive drawing, Fig. 25, AvB^ 
is equal in length to hB of Fig. 24, and Aj,h is easily found, equal 
to Ab of Fig. 24. These lines may be laid off at any convenient 



36 Engin^eertng Descriptive Geometry 

place as the sides of a right triangle, and the hypotenuse measured 
to give the true length of AB. Mathematically the hypotenuse is 
the square root of the sum of the squares of the sides. In the case 
illustrated ArBv is 5 (itself the square root of AvC +BvC ', or 
VS^Ti^ ) and Anl is 3. The length AB is therefore VS' + S^ 
= V34 = 5.83. 

Problems III. 

(For use with wire-mesh cage, cross-section paper, or blackboard.) 

30. A square in a position similar to the pentagon of Fig. 
26 has the corners A (10,13,2), B (2,12,8), C (2,2.8) and 
D (10, 2, 2). Find its true shape by the use of an auxiliary plane. 

31. A square is in a position similar to the pentagon of Fig. 23. 
The comers are A (9,3,3), B (9,13,3), C (3, 13, 11), and 
D (3,3,11). Find its true shape by revolving into a plane par- 
alel to H- 

32. Plot the triangle A (11,3,2), B (12, 6, 12), C (14, 12, 7). 
Find its true shape by the use of an auxiliary plane perpendicular 

to n. 

33. Plot the triangle .4 (13,15,8), B (10,11,0), C (7,7,8). 
Find the true shape of the triangle by revolving it about AAji 
until in a plane parallel to S- Find the true shape by projection 
on a plane \], perpendicular to ff, whose inclined trace passes 
through the point (16,0,0). ("With the wire-mesh cage turn the 
plane S' to serve for this auxiliary plane \].) 

34. Same with triangle A (9, 7, 8), B (12, 11, 13), C (15, 15, 2). 

35. Plot the right triangle A (14, 4, 3), 5 (14, 10, 3), C (6, 4, 9). 
Eevolve it about BBv into a plane parallel to ff and project its 
true shape on Jif. ("With the wire-mesh cage put markers at points 
A, B, C and C", the new position of C.) 

36. Plot the right triangle A (9, 3, 6), B (9, 3, 0), C (15, 11, 6). 
Pevolve it about AB until in a plane parallel to V and plot C", the 
new position of the vertex. Eevolve it about the same axis into a 
plane parallel to S? and plot C", the new position of the vertex. 
(With the wire-mesh cage put point markers at A, C, C and C".) 



The True Length of a Lixe ix Space 37 

37. Plot the square A (14,8,2), B (ll,2,Ti), C (ll,14,7i), 
D (8, 8, 12f ). The diagonal is 12 -units long. Eevolve the square 
about A At into a plane parallel to Ji, and project its true shape on 
ff. (With the wire-mesh cage put point markers at A, B, C, D, 
B', C, and D'.) 

38. Plot the triangle A (12,2,14), B (2,2,14), C (7,7,2). 
Eevolve it about A B into a plane parallel to \, and project the true 
shape on V- (With the wire-mesh cage put markers at points 
A, B, C and C". On coordinate paper or blackboard show the true 
shape b}' projection on an auxiliar}^ plane IJJ perpendicular to S? 
through the point (0, 8, 0).) 

(For use on coordinate paper or blackboard, not wire-mesh cage.) 

39. The triangle A (3,7,11), B (13,2,13), C (5,2,1) is a 
triangle in an oblique plane. Find its true shape as follows : BC 
appears at its true length in V- Draw AvD^ perpendicular to Bvd: 
AD is an oblique line, but it is perpendicular to BC since its V 
projector-plane AAvDvD is perpendicular to BC. Find the true 
length of AD hj any method. On V extend AyDv to Ev, making 
Di-Ev equal to the true length of AD. EvBvCv is the true shape of 
the triano-le ABC. 



CHAPTER IV. 

PLANE SURFACES AND THEIR INTERSECTIONS AND 
DEVELOPMENTS. 

36. The Omission of the Subscripts, h, v, and s — In a descriptive 
drawing a point does not itself appear but is represented by its 
projections on the reference planes. This fact has been emphasized 
in the previous chapters. In the more complicated drawings which 
now follow it will save time and will prevent overloading the figures 
with lettering, to omit the subscripts h, v, and s, and to refer to a 
point and its projections by the same letter. Thus " Av " or " the 
point A in V " ^.re expressions which call attention to the projec- 
tion of A on Vj but a diagram will show only the letter A at that 
place. If at any time it is necessary to be more precise the sub- 
scripts may be restored. They should be used if any confusion is 
caused by their omission. 

If the projections of two points coincide, it is sometimes advis- 
able to indicate which point is behind the other in that view by 
forming the letter of fine dots. "Referring back to Fig. 16, the 
projections of A and B on S coincide. On this system subscripts 
are omitted and the letter 5 (on S only) is formed of dots, 
as in Fig. 27. 

37. Intersecting Plane Faces. — Many pieces of machines and 
structures which form the subjects of mechanical drawings, are 
pieces all of whose surfaces are portions of planes, each portion or 
face having a polygonal outline. 

In making such drawings there arise problems as to the exact 
points and lines of intersection, which can be solved by applying 
the laws of projection treated of in the preceding chapters. How 
these intersections are determined from the usual data will now 
be shown. 

38. A Pyramid Cut by a Plane. — As a simple example let us 
suppose that it is required to find where a plane perpendicular to 
V, and inclined at an angle of 30° with W, intersects an hex- 



Plane Surfaces and Their Intersections 



39 



agonal pyramid with axis perpendicular to ff. Fig, 27 is the 
drawing of the pyramid, having the base ABCDEF and vertex P. 
The cutting plane is an inclined plane such as we have used for an 
auxiliary plane, and its traces are therefore similar to those of an 
auxiliaTy plane. KL is the inclined trace on V and KK' and LL' 
are the traces parallel to the axis of Y. The problem is to find the 
shape of the polygonal intersection abode f in f\ and S, and its 
true shape. 




The method of solution of all such problems is to take into con- 
sideration each edge of the pyramid in turn, and to trace the points 
where they pierce the plane. Thus, the edge PA pierces the given 
plane at a, whose projection on V is first located; for the given 
plane is seen on edge in \, and PA cannot pierce the plane at any 
other point consistent with that condition, a, once located in V> 
can be projected horizontally to the line PA in S and vertically to 
PA in H. 

The true shape of the polygon ahcdef may be shown on an aux- 
iliary plane, U, whose traces are ZM and ZN. In Fig. 27 the 
projection of the pyramid on U is incomplete. As it is only to 
show the polygon ahcdef the rest of the figure is omitted. 



40 



Engineerin^g DEScrxiPTivE Geometry 



39. Intersecting Prisms. — As an example of somewhat greater 
diflticulty let it be required to find the intersection of two prisms, 
one, the larger, having a pentagonal base, parallel to lil ; and the 
other a triangular base, parallel to S- The axes intersect at right 
angles, and the smaller prism pierces tlie larger. 




X , 

If 



V 



z 

Fig. 28. 




Y« X I 

E D V iA C B 



^ 



4 



Y' e' 



a- 



At 



E!^T 



V 



Li 



I 






Way 



WF 



z 

Fig. 29. 



A i-J 


B 


P D 


C 




'^ 


..rt 


^ 


.1 




F 


b' 


H 




; 


i i-' 


B' 


• V 


c 



S i 



The known elements or data of the problem are shown recorded 
as a descriptive drawing in Fig. 28. It shoAvs the projection of 
the pentagon on ff, of the triangle on g, and of the axes on V- 
The problem is to complete the drawing to the condition of Fig. 
29, shown on a larg^er scale. The corners of the pentagonal prism 
are ABCDEF and A'B'C'D'E'F' and its axis is PF. The corners 
of the triangular prism are FGH and F'G'H' and its axis is QQ' . 

40. Points of Intersection. — The general course in solving the 
problem of the intersection of the prisms is to consider each edge 
of each prism in turn, and to trace out where each edge pierces the 
various plane faces of the other prism. When all such points of 



Plane Surfaces and Their Intersections 41 

intersection have been determined, they are joined by lines to rive 
the complete line of intersection of the prisms. 

To determine where a given edge of one prism cuts a given plane 
face of the other prism, that view in which the given plane face is 
seen as a line only, or is " seen on edge," as is said, mnst be re- 
ferred to. Taking the hexagonal prism first, the edges AA', CC, 
and DD' entirely clear the triangular prism, as is disclosed by the 
plan view on ff where they appear '' on end " or as single points 
only. They, therefore, have no points of intersection with the 
triangular prism and in V and S these lines may be drawn as 
uninterrupted lines, being made full or broken according to the 
rule at the end of Art. 5. BB', as may be seen in U, meets the 
small prism. This line when drawn in S, where the plane faces 
FF'G'G and F FIT II are seen on edge, meets those faces at 6 and 
V. From S these points are projected to V- The edge BB' con- 
sists really of two parts, Bh and VB'. EE' meets the same two 
faces at points e and e' determined in the same way. 

FF', when drawn in H, is seen to pierce the plane face AA'B'B 
at / and AA'E'E at f. These points, located in H, are projected 
vertically down to V- GG' in H pierces BB'C'C at g, and EE'D'D 
at g'. h and h' on the line IIH' are similarly determined first in 
fi and are projected down to V- 

41. Lines of Intersection.— Having found the points of inter- 
section of the edges, we determine the lines of intersection of the 
plane surfaces by considering the intersections of plane with plane, 
instead of Ime with plane. BB' is one line of the plane AA'B'B, 
and pierces the plane FF'G'G (seen on edge in S) at &. b is there- 
fore a point of both planes. FF' is a line of the plane FF'G'G, and 
it pierces the plane AA'B'B (seen on edge in H) at /.• f is also a 
point common to both planes. Since these two points are in both 
planes, they are points on the line of intersection of the two planes. 
We therefore connect b and / by a straight line in V, but do not 
extend it beyond either point because the planes are themselves 
limited. 

By the same kind of reasoning & and ^ are found to be points 
common to BB'C'C and FF'G'G, and are therefore joined by a 
straight line, bg in V- gli also is the line of intersection of two 



42 



Engineeeing Descriptive Geometry 



planes, and the student should follow for himself the full process 
of reasoning which proves it, e, f, and g' are points similar to h, f, 
and g. Since the original statement required the triangular prism 
to pierce the pentagonal one, gg', ff, and hh' are joined by broken 
lines representing the concealed portions of the edges GG', FF', and 
HH' of the small prism. Had it been stated that the object was 
one solid piece instead of two pieces, these lines would not exist on 
the descriptive drawing. 

42. Use of an Auxiliary Plane of Projection. — To find the inter- 
section of solids composed of plane faces, it is essential to have 





D 


C 




^ 




}- 


~A 


A 




b 


/J 


. 1 1 




e' 




">~>^ \i| 




A 1 ^ 


''■^ 


1 

1 


f"-^ 


/j'b 






Fig. 30. 



Fig. 31. 



views in which the various plane faces are seen on edge. To obtain 
such views, an auxiliary plane of projection is often needed. 

Fig. 30 shows the data of a problem which requires the auxiliary 
view on U in order to show the side planes of the triangular prism 
"on edge." (These planes are oblique, not inclined, and therefore 
do not appear "on edge" on any reference plane.) Fig. 31 shows 
the completie solution, the object drawn being one solid piece and 
not one prism piercing another prism, h and d are located by the 
use of the view on \]. In this case and in many similar cases in 
practical drawing, the complete view on \J need not be constructed. 



Plane Surfaces and Their Intersections 



43 



The use of U is only to give the position of & and d, which are then 
projected to V- The construction on \J of the square ends of the 
square prism are quite superfluous, and would be omitted in prac- 
tice. In fact, the view HJ would be only partially constructed in 
pencil, and would not appear on the finished drawing in ink. 

After the method is well understood, there will be no uncertainty 
as to how much to omit. 

43. A Cross-Section. — In practical drawing it often occurs that 
useful information about a piece can be given by imagining it cut 
by a plane surface, and the shape of this plane intersection drawn. 
In machine drawing, such a section showing only the material 
actually cut by the plane and nothing beyond, is called a " cross- 





H 


c 


3* 


X 




1 

, 1 








~1 




1 ' 




m. 




c 


J ' 1 


n 


















X 







Z 

32. 




1 


If 


1 




m 


- 


V 




Fig. 



/ 


1 
1 
1 


1 


l\ 


f 11 





s 



section." In other branches of drawing other names for the same 
kind of a section are used. The " contour lines " of a map are of 
this nature, as well as the " water lines " of a hull drawing in 
iSTaval Architecture. 

44. Sectional Views. — The cross-section is used freely in ma- 
chinery drawing, but a " sectional view," which is a view of a cross- 
section, with all those parts of the piece which lie beyond the plane 
of the section as well, is much more common. 

These sectional views are sometimes made additional to the 
regular views, but often replace them to some extent. Fig. 32 is a 



'44 Engineering Descriptive Geometry 

.good example. It represents a cast-iron structural piece shown by 
plan, and two sectional views. The laws of projection are not 
^altered, but the views bear no relation to each other in one respect. 
One view is of the whole piece, one is of half the piece, and one is 
of three-quarters of it. The amount of the object imagined to be 
■cut away and discarded in each view is a matter of independent 
choice. 

In the example the projection on V is a view of half of the 
piece, imagining it to have been cut on a plane shown in ff by the 
line mn. The half between mn and OX has been discarded, and the 
drawing shows the far half. The actual section, the cross-section 
on the line m7i, is an imaginary surface, not a true surface of the 
object, and it is made distinctive by "hatching.^' This hatching 
is a conventional grouping of lines which show also the material 
of which the piece is formed. For this subject, consult tables of 
standards as given in works on Mechanical Drawing. This pro- 
jection on V is not called a " Front Elevation," but a " Front Ele- 
vation in Section," or a " Section on the Front Elevation." 

The view projected on S is called a " Side Elevation, Half in 
Section," or a " Half-Section on the Side Elevation." Since a 
section generally means a sectional view of the object with Jialf 
removed, a half-section means a view of the object with one-quarter 
removed. If, in ff, the object is cut by a plane whose trace is np 
^nd another whose trace is pq, and the IST. E. comer of the object 
is removed, it will correspond to the condition of the object as seen 
in S. 

Sections are usually made on the center lines, or rather on central 
planes of the object. When strengthening ribs or " webs " are seen 
in machine parts, it is usiial to take the plane of the section just 
in front of the rib rather than to cut a rib or web which lies on the 
central plane itself. This position of the imaginary saw-cut is 
selected rather than the adjacent center line. 

When the .plane of a section is not on a center line, or adjacent 
to one, its exact location should be marked in one of the views in 
which it appears " on edge," and reference letters put at the ex- • 
tremities. The section is then called the " section on the line mn.'^ 

The passing of these section planes causes problems in intersec- 
tion to arise, which are similar to those treated in Articles 37-43. 



Plane Surfaces and Their Intersections 



45 



45. Development of a Prism. — It is often desired to show the 
true shape of all the plane faces of a solid object in one view, 
keeping the adjacent faces in contact as much as possible. This 
is called developing the surface on a plane, and is particularly 
useful for all objects made of sheet-metal, as the development forms 
a pattern for cutting the metal, which then requires only to be bent 
into shape and the edges to be joined or soldered. 




Development is a process already applied to the planes of pro- 
jection themselves when these planes were revolved about axes imtil 
all coincided in one plane. The same operation applied to the 
surfaces of the solid itself produces the development. 

The two prisms of Fig. 29 afford good subjects for development. 
Fig. 33 shows the developed surface of the triangular prism, the 
lines g-g and g'-g' showing the lines of intersection with the other 
prism. In this figure it is considered that the surface of the 
triangular prism is cut along the lines GG', GF, G'F, GH, and 



46 Engineeeixg Descriptive Geometry 

G'H' ; and the four outer planes unfolded, using the lines bound- 
ing FF'H'H as axes, until the entire surface is flattened out on the 
plane of FF'H'H. 

Fig. 34 shows the development of the large prism of Fig. 29, 
with the holes where the triangular prism pierces it when the two 
are assembled. The surface of the prism is cut on the line AA', 
and on other lines as needed, and the surfaces are flattened out by 
unfolding on the edges not cut. 

The construction of these developments is simple, since the sur- 
faces are all triangles or pentagons whose true shapes are given; 
or are rectangles, the true length of whose edges are already known. 

In Fig. 33 the distances Gg, G'g', Ff, F'f are taken directly 
from V in Fig. 29. The points & and e are plotted as follows: 
The perpendicular distance hi to the line GF is taken from \, 
Fig. 29, and Gl is taken from Gl in Sj Fig. 29. The other points 
are plotted in the same manner. 

46. Development of a Pyramid. — Fig. 35 shows the development 
of the point of the pyramid, Fig. 27, cut off by the intersecting 




plane whose trace is KL. The base is taken from the projection 
on HJ, where its true shape is given. Each slant side must have its 
true shape determined, either as a whole plane figure (Art. 31), 
or by having all three edges separately determined (Art. 28 or 
Art. 32). In this case Pa and Pd are shown in true length in V-. 
Fig. 27, and it is only necessary to determine the true lengths of 
Ph and Pc (or their equivalents Pf and Pe) to have at hand all the 
data for laying out the development. The face Pef may be con- 
veniently shown in its true shape on an auxiliary plane "W? Fig. 27, 
perpendicular to S and cutting S in a trace YsN as shown. 



Plane Surfaces and Their Intersections 47 

Problems IV. 

(For use with wire-mesh cage, or on cross-section paper or 
blackboard. ) 

40. Plot the projections of the points A (9, 3, 16), B (6, 3, 16), 
C (6, 8, 16), D (9, 8, 16), and E (0, 3, 4), F (0, 3, 8), G (0, 8, 8), 
H (0, 8, 4). Join the projections, A to E, B to F, C to G, etc. 
(With the wire-mesh cage use stiff wire to represent the lines AE, 
BF, etc.) Show how to find the true shape of every plane surface 
of the solid (a prism) thus formed. On cross-section paper or on 
blackboard show how to draw the development of the surface of 
the solid. 

41. Same as Problem 40, with points A (10, 8, 0), 5 (8, 10, 0), 
C (12, 14, 0), 7? (14,13,0) on H and i^ (10, 8,16), F (6,12,16), 
G (8, 14, 16), H (12, 10, 16) on H'. (In developing the surface, 
find the true shape of the quadrilateral BFGC by dividing it into 
two triangles by a diagonal BG whose true length will appear on S- 
Divide CGHD by the diagonal CH.) 

42. Draw the tetrahedron whose four comers are A (16, 2, 13), 
B (6, 2, 13), C (11, 14, 13) and D (11, 7. 1). It is intersected by 
a plane perpendicular to V cutting V in a trace passing through 
the origin, making an angle of 30° with OX. Draw the trace of 
the plane on \. Where are its traces on W and S ? Show the f\ 
and S projections of the intersection of the plane and tetrahedron. 

43. A solid is in the form of a pyramid whose base is a square of 
10 units, and whose height is 8 units. The corners are A ( 16, 2, 10) , 
B (10,2,2), C (2,2,8) and D (8,2,16) and the vertex 
E (9,10,9). It is intersected by a plane perpendicular to f\, 
whose trace on W passes through the origin making an angle of 
30° with OX. Draw the V and S projections of the intersection 
of the pyramid and plane. Where is the trace of the cutting plane 
on V? 

44. A plane W is parallel to H at a distance of 16 units. A 
square prism has its base in ff- The corners are A (8,2,0), 
B (3, 7, 0), C (8, 12, 0), D (13, 7, 0). Its other base is in H', the 
corners A', B', etc., having the same x and y coordinates as above, 
and the z coordinates 16. 



48 Engineering Descriptive Geometry 

A plane S' is parallel to S at a distance of 16. A triangular prism 
has its base in g, points E (0, 5, 8), F (0, 13, 2), G (0, 13, U) ; 
and its other base in §', points E', F', G' having x coordinates 16, 
and y and z coordinates -unchanged. Make the drawing of the in- 
tersecting prisms considering the triangular prism to be solid and 
parts of the square prism cut away to permit the triangular one to 
pass through. 

(For use on cross-section paper or blackboard, not wire-mesh 
cage.) 

45. A sheet-iron coal chute connects a square port, A (2,4,2), 
B (2,13,2), C (2,12,10), D (2,4,10), with a square hatch, 
E (14, 6, 16), F (14, 10, 16), G (10, 10, 16), H (10, 6, 16). The 
corners form lines AE, BF, CG, DH and the side plates are bent 
on the lines AH and BG. Draw the development of the surface. 

.46. Draw the development of the tetrahedron of Problem 42 
with the line of intersection marked on it. 

47. Draw the development of the pyramid of Problem 43 with 
the line of intersection marked on it. 

48. Draw the development of the square prism of Problem 44 
with the line of intersection marked on it. 

49. Draw the development of the triangular prism of Problem 
44 with the line of intersection marked on it. 



CHAPTEK V. 

CURVED LINES. 

47. The Simplest Plane Curve, the Circle. — The geometrical 
natures of the common curves are supposed to be understood. De- 
scriptive Geometry treats of the nature of their orthographic pro- 
jections. The curves now considered are plane curves, that is, 
every point of the curve lies in the same plane. It is natural, 
therefore, that the relation of the plane of the curve to the plane of 
projection governs the nature of the projection. 




Fig. 36. 



Fig. 37. 



The simplest plane curve is a circle. Figs. 36 and 37 show the 
three forms in which it projects upon a plane. In Fig. 36, a per- 
spective drawing, we have a circle projected upon a parallel plane 
of projection (that in the position customary for V)- The pro- 
jectors are of equal length and the projection is itself a circle ex- 
actly equal to the given circle. 

On a second plane of projection (that in the position of S) per- 
pendicular to the plane of the circle the projection is a straight 
line equal in length to the diameter of the circle, AC. The pro- 
jectors for this second plane of projection form a projector-plane. 



50 



Engineerijstg Descriptive Geometry 



In Fig. 37 the circle is in a plane inclined at an angle to the 
plane of projection. The projectors are of varying lengths. There 
must be one diameter of the circle, however, that marked AC^ 
which is parallel to the plane of projection. The projectors from 
these points are of equal length, and the diameter AC appears of 
its true length on the projection as AvCv. 

The diameter BD at right angles to AG. has at its extremity B 
the shortest projector, and at the extremity D the longest projector. 
On the projection, BD appears greatly foreshortened as BvD„, 
though still at right angles to, the projection of AC and bisected 
by it. ,,v} 




The true shape of the projection is an ellipse, of which AvCv is 
the major axis and BvDv is the minor axis. No matter at what 
angle the plane of projection lies, the projection of a circle is an 
ellipse whose major axis is equal to the diameter of the circle. 

Eor convenience the two planes of projection in Fig. 36 have 
been considered as V ^^^ S? and the projections lettered accord- 
ingly. The plane of projection in Fig. 37 has been treated as if 
it were V? an*! the ellipse so lettered. It must be remembered that 
the three forms in which the circle projects upon a plane, as a 
circle, as a line, and as an ellipse, cover all possible cases, and the 
relations between the plane of the circle and the plane of projec- 
tion shown in the two figures are intended to be perfectly general 
and not confined to V ai^i S alone. 



Curved Lines 



51 



48. The Circle in a Horizontal or Vertical Plane. — Passing now 
to the descriptive drawing of a circle, the simplest case is that of 
a circle which lies in a plane parallel to ]M, V or S- The projec- 
tions are then of the kind shown in Fig. 36, two projections being 
lines and one the true shape of the circle. Fig. 38 shows the case 
for a circle lying in a horizontal plane. The true shape appears in 
H. The V projection shows the diameter AG, the. S projection 
shows the diameter BD. 




Fig. 39. 



49. The Circle in an Inclined Plane. — Fig. 39 shows the circle 
lying in an inclined plane, perpendicular to V? aii<i making an 
angle of 60° with lil. The V projectors, lying in the plane of the 
circle itself, form a projector-plane and the V projection is a 
straight line equal to a diameter of the circle. As the plane of the 
circle is oblique to ff and S, these projections on ff and S are 
ellipses whose major axes are equal to the diameter of the circle. 
Of course, for any point of the curve, as P, the laws of projection 
hold, as is indicated. The true shape of the curve can be shown by 



53 



Engineering Descriptive Geometry 



projection on any plane parallel to the plane of the circle. It is 
here shown on the auxiliary plane \], taken as required. If the 
drawing Avcre presented with projections fi, V and S? as shown, 
one might at first suspect that it represented an ellipse and not a 
circle ; but, if a number of points were plotted on \], the existence 
of a center 0' could be proved by actual test with the dividers. 

50. The Circle in an Oblique Plane. — ^^Vhen a circle ]s in an 
oblique plane, all three projections are ellipses, as in Fig. 40. The 
noticeable feature is that the three major axes are all equal in 
lencrth. 




When an ellipse is in an oblique plane, its three projections are 
also ellipses, but the major axes will be of unequal lengihs. The 
proof of this fact must be left until later. The fact that the three 
projections have their major axes equal must be taken at present as 
sufficient evidence that the curve itself is a circle. 

51. The Ellipse: Approximate Representation. — The ellipse is 
little used as a shape for machine parts. It appears in drawings 
chiefly as the projection of a circle. Some properties of ellipses 
are very useful and should be studied for the sake of reducing the 
labor of executing drawings in which ellipses appear. 

An approximation to a true ellipse by circular arcs, known as the 
" draftsman's ellipse," may be constructed Avhen the major axis 2a 
and the minor axis 2h, Fig. -il, of an ellipse are kno-wii. 



Curved Lixes 



53 



The steps in the process are shown in Fig. 41. The center of the 
ellipse is at 0. The major axis is AC, equal to 2a. The minor 
axis is DB, equal to 26. From C, one end of the major axis, lay 
off CE, equal to 1). The point jG" is at a distance equal to a — & from 
and at a distance equal to 2a — h from A. This last distance is 
the radius of a circular arc which is used to approximate to the 
flat sides of the ellipse. It may be called the " side arc." Setting 
the compass to the distance AE and using i) and B as centers, 
points H and G are marked on the minor axis, extended, for use 
as centers for the " side arcs.'' These arcs are now drawn (passing 
through the points D and B), as shown in the 2nd stage of the 
process. 




sf- Stage 



End Stage 



3rd Stage 



THE " DRAFTSMAN'S ELLIPSE." 
Fig, 41. 



By use of the bow spacer, the ilistance OE is bisected ^ and the 
half added to itself, giving the point F (distant f (a — h) from 0). 
F is the center of a circular arc which approximates to the end of 
the true ellipse. With F as center, and FC as radius, describe this 
arc. If this work is accurate, this " end arc " will prove to be 
tangent to the side arcs already drawn, as shov/n in the 3rd stage 
of the process. If desired, the exact point of tangency of the two 
arcs, K, may be found by joining the centers TI and F and extend- 
ing the line to K. F is swung about as center by compass or 
dividers to F', for the center of the other " end arc." In inking 
such an ellipse, the arcs must be terminated exactly at the points 
of tangency, K and the three similar points. 

This method is remarkably accurate for ellipses whose minor 
5 



54 



Engineering Descriptive Geometry 



axes are at least two-thirds the lengtli of their major axes. It 
should always be used for such wide ellipses, and if the character 
of the drawing does not require great accuracy, it may be used 
even when the minor axis is but half the length of the majop axis. 
For all narrow ellipses, exact methods of plotting should be used. 
52. The Ellipse: Exact Representation. — The true and accurate 
methods of plotting an ellipse are shown in Figs. 42, 43, and 44. 
Fig. 42 is a convenient method when the major axis AC and minor 
axis BD are given, bisecting each other at 0. Describe circles with 
centers at 0, and with diameters equal to J.C and BD. From 
draw any radial line. From the point where this radial line meets 
the larger circle draw a vertical line, and from the point where it 
cuts the smaller circle draw a horizontal line. Where these lines 




meet at P is located a point on the ellipse. By passing a large 
number of such radial lines sufficient points may be found between 
D and C to fully determine the quadrant of the ellipse. Having 
determined one quadrant, it is generally possible to transfer the 
curve by the pearwood curves with less labor than to plot each 
quadrant. 

With the same data a second method, Fig. 43, is more convenient 
for work on a large scale when the T-square, beam compass, etc., 
are not available. 

Construct a rectangle using the given major and minor axes as 
center lines. Divide DE into any number of equal parts (as here 
shown, 4 parts), and join these points of division with 0. Divide 
DO into the same number of equal parts (here, 4). From A 
.draw lines through these last points of division, extending them to 
the first system of lines intersecting the first of the one system with 



Curved Lines 55 

the first of the other, the second with the second, etc. These inter- 
sections, 1, 2, 3, are points on the ellipse. 

The third method, an extension or generalization of the second, is 
very useful when an ellipse is to be inscribed in a parallelogram, the 
major and minor axes being unknown in direction and magnitude. 
Lettering the parallelogram A'B'C'D' in a manner similar to the 
lettering in Fig. 43, the method is exactly the same as before, D'E' 
and D'O being divided into an equal number of parts and the lines 
drawn from C and A'. The actual major and minor axes, indicated 
in the figure, are not determined in any manner by this process. 

53. The Helix. — The curve in space (not a plane curve) wliich 
is most commonly used in machinery, is the helix. This curve is 
described by a point revolving uniformly about an axis and at the 
same time moving uniformly in the direction of that axis. It is 
popularly called a " cork-screw " curve, or " screw thread," or even, 
quite incorrectly, a " spiral." 

The helix lies entirely on the surface of a cylinder, the radius of 
the cylinder being the distance of the point from the axis of rota- 
tion, and the axis of the cylinder the given direction. 

Fig. 45 represents a cylinder on the surface of which a moving 
point has described a helix. Starting at the top of the cylinder, at 
the point marked 0, the point has moved uniformly completely 
around the cylinder at the same time that it has moved the length 
of the cylinder at a uniform rate. The circumference of the top 
circle of the cylinder has been divided into twelve equal parts by 
radii at angles of 30°, the apparent inequality of the angles being due 
to the perspective of the drawing. The points of division are marked 
from to 11, point 13 not being numbered, as it coincides with 
point 0. The length of the cylinder is divided into twelve equal 
parts on the vertical line showing the numbers from to 12, and 
at each point of division a circle, parallel to the top base, is de- 
scribed about the cylinder. The helix is the curve shown by a 
heavy line. From point 0, which is the zero point of both move- 
ments, the first twelfth part of the motion carries the point from 
to 1 around the circumference, and from to 1 axially downward, 
at the same time. The true movement is diagonally across the 
curved rectangle to the point marked 1 on the helix. This move- 



56 



Engineering DESCRirTivE Geometry 



ment is continued step by step to the points 2, 3, etc. In the posi- 
tion chosen in Fig. 45, points 0, 1, 2, 3, 4, 12 are in full view, 
points 5 and 11 are on the extreme edges, and the intermediate 




Fig. 45. 



points (from G to 10) are on the far side of the cylinder. The 
constrnction linos for these latter points have been omitted, in order 
to keep the fignre clear. 



Curved Lines 57 

54. Projections of the Helix. — Tlie projection of this curve on a 
plane parallel to the axis of the c^-linder is sho^^^l to the loft. The 
circles described about the cylinder become equidistant parallel 
straight lines. The axial lines remain straight but are no longer 
equally spaced, and the curve is a kind of continiious diagonal to 
the small rectangles formed by these lines on the plane of projec- 
tion. 

The projection of the helix on any plane perpendicular to the 
axis of the cylinder is a circle coinciding with the projection of the 
cylinder itself. The top base is such a plane and on it the projec- 
tion of the helix coincides with the circumference of the base, 

55. Descriptive Drawing of the Helix. — The typical descriptive 
drawing of a helix is showTi in Tig. 46. The axis of the cylinder is 
perpendicular to fi, and the top base is parallel to ff. The helix 
in JHI appears as a circle. In V it appears as on the plane of pro- 
jection in Fig. 45, but this view is no longer seen obliquely as is 
there represented. 

This V projection of the helix is a plane curve of such import- 
ance as to receive a separate name. It is called the " sinusoid." 
Since the motion of the describing point is not limited to one com- 
plete revolution, it may continue indefinitely. The part drawn is 
one complete portion and any addition is but the repetition of the 
same moved along the axial length of the curve. The proportions 
of the curve may vaiy hetween wide limits depending on the rela- 
tive size of the radius of the cylinder to the axial movement for one 
revolution. This axial distance is known as the " pitch " of the 
helix. 

In Fig. 46 the pitch is ahout three times the radius of the helix. 
In Fig. 47, a short-pitch helix is represented, the pitch being about 
f the radius, and a number of complete rotations being shown. 

The proportions of the helix depends therefore on the radius and 
on the pitch. To execute a drawing, such as Fig. 46, describe first 
the view of the helix which is a circle. Divide the circumference 
into any number of equal parts (12 or 24 usually). From these 
points of division project lines to the other view or views. Divide 
the pitch into the same number of equal parts, and draw lines per- 
pendicular to those already drawn. Pass a smooth curve through 



58 



Engineeeiistg Descriptive Geometry 



the points of intersection of these lines, forming the continnons 
diagonal. In Figs. 45 and 46 the helix is a "right-hand helix." 
The upper part of Fig. 47 shows a left-hand helix, the motion of 
rotation being reversed, or from 12 to 11 to 10, etc. The ordinary 




■ ' < 


=• S3 


1 


-...==4r-rr^_ 


i 


\ 



Fig. 46. 



Fig. 47. 



screw thread used in machinery is a very short-pitched right-hand 
helix. It is so short indeed that it is customary to represent the 
curve by a straight line passing through those points which would 
be given if the construction were reduced to dividing the circum- 



Curved Lines 



59 



ference and the pitch into 2 equal parts. This is shown in the 
lower part of Fig. 47, where only the points 0, 6 and 12 have been 
used. 

The concealed portion of the helix is then omitted entirely, no 
broken line for the hidden part being allowed by good practice. 

56. The Curved Line in Space. — A curve in space may some- 
times be required, one which follows no known mathematical law, 
but which passes through certain points given by their coordinates. 
For example, in Fig, 48, four points, A (12,1,9), B (5,4,6), 




Fig. 48. 



C (2,4,4) and D (2,5,1), were taken as given and a "smooth 
curve," the most natural and easy curve possible, has been passed 
through them. It is fairly easy to pass smooth curves through the 
projections of the 4 points on each reference plane, but it is essen- 
tial that not only should the original points obey the laws of pro- 
jection of Art. 11, but every intermediate point as well. The 
views must check therefore point by point and the process of trac- 
ing the curve must be carried out about as follows : The projec- 
tions of the 4 points on V a^^d S are seen to be more evenly ex- 
tended than those on ff, and smooth curves are made to pass 
through them by careful fitting with the draftsman's curves. The 



60 Engineeeixg Descriptive Geometey 

view on ff cannot now be put in at random, but must be constructed 
to correspond to the other views. To fill in the wide gap between 
Ah and Bn an intermediate point is taken, as Ev on AvB^. By a 
horizontal line Eg is defined. From Ev and Eg the H projection 
(Eh) ,is plotted by the regular method of checking the projections 
of a point. As many such intermediate points may be taken as may 
seem necessary in each case. 

To define the sharp turn on the curve between Ch and Dh, one 
or more extra points, as Fh, should be plotted from the V and S 
projections. Thus every poorly defined part is made definite and 
the views of the line mutually check. The work of " laying out " 
the lines of a ship on the " mold-loft floor " of a shipbuilding plant 
is of this kind, with the exception that the curves are chiefly plane 
curves, not curves in space. 

Problems V. 

(For blackboard or cross-section paper.) 

50. Make the descriptive drawing of a circle lying in a plane 
parallel to S? center at C (3, 6, 7) and radius 5. 

51. Make the descriptive drawing of a circle lying in a plane 
perpendicular to V and making an angle of 45° with |HI (the trace 
in V passing through the points (18, 0, 0), and (0, 0, 18)). The 
center of the circle is at C (9, 6, 9), and the radius is 5. (Make 
the V projection first, then a projection on an auxiliary plane HJ. 
From these views construct the ff and S projections, using 8 or 9 
points. 

52. Make the descriptive drawing of a circle in a plane perpen- 
dicular to H, the trace in li passing through the points (12, 0, 0) 
and (0,16,0). The center is at (6,8,10) and the radius is 8. 
(Draw the plan and an auxiliary view showing true shape first, and 
from those views construct the projections on V and S.) 

53. An ellipse lies in a plane passing through the axis of Y and 
making angles of 45° with H and g. The H projection is a circle, 
center at (10, 10, 0) and radius 8. Prove that the S projection is 
also a circle and find the true shape of the ellipse by revolving the 
plane of the ellipse into the plane Hi. 



Curved Lines 61- 

54. An ellipse lies in a plane passing through the axis of Y and' 
making an angle of 60° with H and 30° with S- The H projec- 
tion is a circle, center at (8, 8, 0), radius 6. Find the true shapes 
of the ellipse. Construct the view on g by projecting points for the 
center and for the extremities of the axes of the ellipse. Pass a 
draftsman's ellipse through those points. Show that no appreci- 
able error can be observed, 

55. Construct a draftsman's ellipse, on accurate cross-section or 
coordinate paper, with major axis 24 units, and minor axis 13 units. 
Perform the accurate plotting of the true ellipse on the same axes, 
one quadrant by the method of Fig. 43 and one by the method of 
Fig. 43, using 6 divisions for DE and OD. Note the degree of 
accuracy of the approximate process. 

56. On coordinate paper, plot an ellipse by the method of Fig. 
43, the major axis being 16 units long and the minor axis 8 units. 
Plot another ellipse whose major axis is 16 and whose minor axis 
is 13. (To divide the semi-minor axis of 6 units into 4 equal parts, 
use points of division on the vertical line CE instead of OD. CE 
being twice as far from A as OD, 13 units must be used for the 
whole length, and these divided into 4 parts.) 

57. On isometric paper pick out a rhombus like the top of Fig. 
19, but having 8 units on each side. Inscribe an ellipse by plotting 
by the method of Fig. 44. 

- 58. Make the descriptive drawing of a helix whose axis is per- 
pendicular to S through the point (0,7,7). The pitch of the 
helix is 13, and the initial point is (3, 7, 3). Draw the fi and V 
projections of a right-hand helix, numbering tlie points in logical 
order. 

59. Connect the 4 points A (10, 8, 10), B (8, 10, 6), C (6, 9, 4) 
and D (3, 3, 4) by a smooth curve, filling out poorly defined por- 
tions in S from the H and Y projections. 



CHAPTER VI. 
CURVED SURFACES AND THEIR ELEMENTS. 

57. Lines Representing Curved Surfaces, — To represent solids 
having curved surfaces, it is not enough to represent the actual 
corners or edges only. Hitherto only edges have appeared on de- 
scriptive drawings, and it has been a feature of the drawings that 
every point represented on one projection must be represented on 
the other projections, the relation between projections being strictly 
according to rule. We now come to a class of lines which do not 
appear on all tliree. views, lines due to the curvature of the surfaces. 

The general principle, called the " Principle of Tangent Projec- 
tors," governing this new class of lines is as follows : In projecting 
a curved surface to a given plane of projection (by perpendicular 
projectors, of course) all points, and only those points, whose pro- 
jectors are tangent to the curved surface should be projected. A 
good illustration of this principle is shown in Fig. 45, where the 
cylinder is projected upon the plane of projection. The top and 
bottom bases are edges, and project under the ordinary rules, but 
along the straight line 0, 1, 2, . . . . , 12 the curved surface of the 
cylinder is itself perpendicular to the plane of projection. If from 
any point on this line a projector is drawn to the plane of projection 
(as is shown in the figure for the points 1, 3, 3, etc.), this projector 
is tangent to the cylinder. The whole line therefore projects to 
the plane of projection. The projection of the cylinder on a plane 
parallel to its axis is therefore a rectangle, two of its sides repre- 
senting the circular bases and the two other edges representing the 
curved sides of the cylinder. 

58, The Right Circular Cylinder, — The complete descriptive 
drawing of a cylinder is therefore as shown in Fig. 49. This cylin- 
der is a right circular cylinder. Mathematicians consider that the 
cylinder is " generated " by revolving the line AA' about PF, the 
axis of the cylinder. The generating line in any particular posi- 



Curved Surfaces and Their Elements 



63 



tion is called an " element " of the surface. Thus AA\ BB', CC, 
etc., are elements. 

When the cylinder is projected upon V, AA' and CC are the 
elements which appear in V because the V projectors of all points 
along those lines are tangent to the cylinder, as can he seen from 
the view on ff. The elements which are represented by lines on 
S are BB' and DD'. 

The right circular cylinder may also be considered as generated 
by moving a circle along an axis perpendicular to its own plane 
throusrh its center. 




In Fig. 45 consider the top base of the cylinder to be moved 
down the cylinder. Each successive position of the circle is a " cir- 
cular element" of the cylinder. The circles through the points 
1, 2, 3, etc., are simply circular elements of the cylinder taken at 
equal distances apart. , 

59. The Inclined Circular Cylinder. — Eig. 50 shows an inclined 
circular cylinder. It has circular and straight line elements as 
before, though it cannot be generated by revolving a line about 
another at a fixed distance, but can be generated by moving the 
circle ABCD obliquely to- A'B'C'D', the center moving on the axis 
PP'. The straight elements are all parallel to the axis. The cross- 
section of a cylinder is a section taken perpendicular to the axis. 



64 



Engineering Descriptive Geometry 



In this case the cross-section is an ellipse, and for this reason the 
Inclined Circular Cylinder is sometimes called the Elliptical Cyl- 
inder. 

60. Straight and Inclined Circular Cones. — If a generating line 
AP, Eig. 51, meets an axis PP' at a point P, and is revolved about 
it, it will generate a Straight Circular Cone. The cone has both 
straight and circular elements, the circular elements increasing in 
size as they recede from the vertex P. The base ABCD is one of 
the elements. 




The Inclined Circular Cone (Fig. 52) has straight and circular 
elements, but it is not generated by revolving a line about the axis. 
The circular elements move obliquely along the axis PP' and in- 
crease uniformly as they recede from the vertex P. 

61. The Sphere. — ^The Sphere can be generated by revolving a 
semicircle about a diameter. Each point generates a circle, the 
radii of the circles for successive points having values varying 
between and the radius of the sphere. Since the sphere can be 
generated by using any diameter as an axis, the number of ways in 
which the surface can be divided into circular elements is infinite. 

62. Surfaces of Revolution. — In general, any line, straight or 
curved, may be revolved about an axis, thus creating a surface of 
revolution. Every point on the " generating line " creates a " cir- 



Curved Surfaces and Their Elements 



65 



ciilar element" of the surface, and the plane of each circular ele- 
ment is perpendicular to the axis of the surface. 

The straight circular cylinder is a simple case of the general 
class of surfaces of revolution. To generate it a straight line is 
revolved about a parallel straight line. The different points of the 
generating line create the circular elements of the cylinder, and 





Fig. 54. 



the different positions of the generating line mark the straight ele- 
ments. The cone and the sphere are also surfaces of revolution, as 
they are generated by revolving a line about an axis. 

If, a circle be revolved about an axis in its own plane, but en- 
tirely exterior to the circle, a solid, called an " anchor ring," is 
generated. A small portion of this surface, part of its inner surface, 
is often spoken of as a " bell-shaped surface," from its similarity 
to the flaring edge of a bell. 

Any curved line may create a surface of revolution, but in de- 



'6Q Engineering Descriptive Geometry 

signs of machinery lines made up of parts of circles and straight 
lines are most frequently used. Figs, 53 and 54 show two exam- 
ples which illustrate well the application of the Principle of Tan- 
gent Projectors. The generating line is emphasized and the cen- 
ters of the various arcs are marked. 

Any angular point on the generating line, as a (Fig. 53), creates 
a circular edge on the surface. This edge appears as a circle on the 
plan (as aa' on H), and as a straight line, equal to the diameter, 
on the elevation (as aa' on V)- See also the point Ji (Fig. 54), 
In addition, any portion of the generating line which is perpen- 
dicular to the axis, as h (Fig. 53), even if for an infinitely short 
distance only, creates a line on the side view, as ih' on V? but no 
corresponding circle on IHI. A V projector from any point on the 
circular element created by the point h is tangent to the surface, 
and therefore creates a point on the drawing, but an H projector 
is not tangent to the surface, e is a similar point, and so also is 
j of Fig. 54. 

Any point, as c. Fig. 53, where the generating line is parallel to 
the axis for a finite, or for an infinitely small distance, generates 
a circular element, from every point of which the ff projectors are 
tangent to the surface, but the V projectors are not. A circle cc' 
appears, therefore, on the plan for this element of the surface of 
revolution, but no straight line on the side view. ^ is a similar 
point, as are also / and g, on Fig, 54. 

63. The Helicoidal Surface. — If a line, straight or" curved, is 
made to revolve uniformly about an axis and move uniformly along 
the axis at the same time, every point in the line will generate a 
helix of the same pitch. The surface swept up is called a Heli- 
coidal Surface, 

The generating line chosen is usually a straight line intersecting 
the axis. The surfaces used for screw threads are nearly all of 
this kind. Fig. 55 gives an example of a sharp V-threaded screw, 
the two surfaces of the thread having been generated by lines in- 
clined at an angle of 60° to the axis. Fig. 56 shows a square 
thread, the generating lines of the two helicoidal surfaces being 
perpendicular to the axis. Any particular position of the straight 
line is a " straight element " of the helicoidal surface. 



Curved Surfaces and Their Elements 



67 



64. Elementary Intersections. — In executing drawings of ma- 
chinery it is often necessary to determine the line of ititeisection of 
two surfaces, plane or curved. The simplest lines of intersection 
are such as coincide with elements of a curved surface. They may 




Fig. 55. 



Fig. 56. 



be called " Elementary Intersections." An elementary intersection 
may arise when a curved surface is intersected by a plane, so placed 
as to bear some simple relation to the surface itself. 

In Fig. 49, any plane perpendicular to the axis of the cylinder 
intersects it in a circular element of the cylinder, and any plane 
parallel to the axis of the cylinder (or containing it) intersects 



68 Engineehixg Desceiptive Geometry 

it (if it intersects it at all) in two straight line elements of the 
cylinder. 

In Fig. 50 any plane parallel to the base of the cylinder inter- 
sects it in a circular element, and any plane parallel to the axis, 
or containing it, intersects it in straight elements of the cylinder. 

In Fig. 51 or 53 any plane parallel to the base of the cone inter- 
sects it in a circular element, and any plane containing the vertex 
of the cone (if it intersects at all) intersects the cone in straight 
elements. 

In Fig. 53 or 54 any plane perpendicular to the axis of the sur- 
face of revolution intersects it in a circular element. 

In Fig. 55 or 56 any plane containing the axis of the screw inter- 
sects the helicoidal surfaces in straight elements. The plane per- 
pendicular to ff, cutting H in a trace PQ, and cutting V in ^^ 
trace QR, cuts the helicoidal surfaces at each convolution in straight 
elements. Only ah and a'h are marked on the figure. 

Problems VI. 

(For blackboard or cross-section paper or wire-mesh cage.) 

60. Draw the projections of a cylinder whose axis is P (6, 2, 6), 
P' (6, 16, 6), and radius 5. Draw the intersection of this cylinder 
with a plane parallel to H, at 4 units from ff, and with a plane 
parallel to V^ 10 units from V. 

61. An inclined circular cylinder has its bases parallel to S- Its 
axis is P (2, 7, 7), P' (14, 7, 13). Its radius is 5. Draw the V 
and S projections and the intersection, with a plane parallel to Sr 
6 units from S? and with a plane parallel to V? 3 units from V- 

62. Draw a cone with vertex at P (4,8,8), center of base at 
P' (16, 8, 8), and radius 6, the base lying in a plane parallel to S- 
Draw the intersection with a plane parallel to S> 13 units from S> 
and with a plane perpendicular to S, whose trace in S passes 
through the points (0, 8, 8) and (0, 14, 0). 

63. An oblique cone has its vertex at P (16, 8, 4) and its base in 
a plane parallel to ff, center at P' (8, 8, 16), and radius 5. Draw 
the intersection with a plane parallel to JM^ 1^ units from fi, and 
with a plane containing the axis and the point (16, 0, 16). 



CuEVED Surfaces axd Their Elements 69 

64:. A cone has an axis P (8,2,2) , P' (8, 14, 10). Its base is in 
a plane iDarallel to V, 14 units from V, and its radius is 6 units. 
Draw the intersection with a plane containing the vertex and the 
points (0,14,12) and (16,14,12). 

65. A surface of revolution is formed by revolving a circle, whose 
center is at (12, 8, 8) and radius 3 units, lying in a plane parallel 
to Vj about an axis perpendicular to ff at the point (8, 8, 0). It 
is cut by a plane parallel to li at a distance of 6 units from ff. 
Draw the intersections. 

66. A sphere has its center at (8, 8, 8) and a radius of 5 units. 
Draw the intersection with a cylinder whose axis is P (8, 8, 0), 
P' (8,8,16), and whose radius is 4 units, its bases being planes 
perpendicular to its axis. 

67. A sphere has its center at (8, 8, 8) and a radius of 5 units. 
Find its intersection with a cone whose vertex is P (0, 8, 8), center 
of base (16, 8, 8), and radius of base 6 units, the base being in a 
plane S' parallel to §. 

68. In Fig. 53 let the generating line Pabcde be revolved about 
ee' as an axis, x^ssume any dimensions for the line and draw the 

V and S projections of the surface of revolution thus formed. 
Draw the intersection with a plane parallel to S j^st to the right 
of d. 

69. In Fig. 54 let the generating line Pfgli be revolved about 
lili' as an axis. Assume any dimensions for the line and draw the 

V and 3 projections of the surface of revolution formed. 



CHAPTEE YIL 
INTERSECTIONS OF CURVED SURFACES. 

65. The Method of the Intersection of the Intersections. — The 

determination of the line of intersection of two curved surfaces (or 
of a curved surface and a plane), when not an " Elementary Inter- 
section/' is of much greater difficult)- and requires a clear under- 
standing of the nature of the curved surfaces themselves, and some 
little ingenuity in applying general principles. 

The method generally relied upon for the solution is the use of 
auxiliary intersecting planes so chosen as to ciit elementary inter- 
sections with each of the given surfaces. These elementary inter- 
sections are drawn and the points of intersection of the intersec- 
tions are identified and recorded as points on the required line of 
intersection. This method is spoken of as " finding the intersec- 
tion of the intersections/' When a number of auxiliary planes 
have been used in this way, a smooth curve is passed through the 
points on the required intersection of the surfaces, as described in 
Art. 55. It should not be necessary, however, to interpolate points 
to fill out gaps as was done in Fig. -IS for E and F. This can be 
done better by the use of more auxiliary intersecting planes. Ex- 
amples of tliis method will make it clear. 

66. An Inclined Circular Cylinder Cut by an Inclined Plane. — 
In Fig. 57 an inclined cylinder, axis PP\ is cut by a plane perpen- 
dicular to v.- and inclined to ff. The traces of this plane are IJ 
in U, J^ in V, and Jii in S- 

It is an Inclined Plane (see Art. 19), not an Oblique Plane. 
Having the descriptive drawing of the cylinder and the traces of 
the plane given, the problem is to draw the line of intersection of 
the siirfaces. It is well-known that in this case the line of inter- 
section is an ellipse, but the method of determining it permits the 
ellipse to be plotted whether it is recognized as such or not. No 
use is to be made of previous knowledge of the nature of the curve 



Intersections of Curved Surfaces 



71 



of intersection of any of the cases treated in this and the next 
chapter. 

Two Yariations of the method are applicable in this case. In the 
first method, auxiliary intersecting planes may be taken parallel to 
the axis of the cylinder. The simplest method of doing this is to 
take auxiliary planes parallel to V, since the axis itself is parallel 




to V. Let R'R be the trace on M, and ER" the trace on S of a 
plane parallel to V. AVe may call this plane simply " R." 

Let e and / be the points where R'R cuts the top base of the cyl- 
inder. Project these points from H to V and in V draw ee' and 
ff parallel to PP'. These straight elements of the cylinder are the 
lines of intersection of the auxiliary plane with the cylinder. As 
a check on the work, e' and f, where R'R in H cuts the bottom 
base of the cylinder, should project vertically to e' and /' in V- 



72 Engixeeeixg Descriptive Geometet 

The auxiliary plane ciits the giren plane JK in a line of inter- 
section whose projection on V coincides with JK itself. 

The points j and Tc, where ee' and //' intersect JK, are the " inter- 
sections of fiie intersections," and are therefore points on the line 
cf intersection of the cylinder and the plane K. Project j and A; 
to E'E on W and to EE" on S- These are points on the required 
curves in J-f and S. By extending in W the projecting lines of 
j and h as far above the axis FP' as ;' and A.- are below it, / and V, 
points on the upper half of the cylinder, symmetrical with ;" and A: 
en the lower half, are found. The construction is equivalent to 
passing a second auxiliary plane parallel to PP' at the same dis- 
tance from PP' as E, but on the other side. 

By passing a number of planes similar to E, a sufficient number 
of points are located to define accurately the ellipse ahcd in W 
and S' 

The true shape of this ellipse is shown in U. a plane parallel to 
JK, at any convenient distance. In the example chosen, the plane 
JK has been taken perpendicular to PP', so that the ellipse abed is 
the true cross-section of the cylinder. Xothing in the method de- 
pends on this fact and it is perfectly general and applicable to any 
inclined plane. 

A variation may be made by passing the auxiliary planes per- 
pendicular to V aiid parallel to PP'. ee' in V may be taken as the 
trace of such a plane. The intersections of this auxiliary with both 
surfaces should be traced and the intersection of the intersections 
identified and recorded as a point of the curve required. / and / 
are the points thus found. This method indeed requires the same 
ccnstruction lines as before, but gives a different explanation to 
them. 

67. A Second Method Using Circular Elements of the Cylinder. — 
A plane parallel to the base of the cylinder and therefore, in this 
case, parallel tp Ifil, will cut the cylinder in a line of intersection 
which is one of the circular 'elements of the cylinder. Let T'T and 
TT", in Fig. 58, be the traces of a plane " T" parallel to JM. The 
axis of the cylinder PP' pierces the plane T at /;. p is therefore 
tlie center of the circle of intersection of the auxiliary plane T with 
the cylinder. Project p to H- and using p as a center and with a 
radius equal to pt, describe the circle as shown. 



Intersections of Curved Surfaces 



The planes T and JK are both perpendicular to V or " seen on 
edge " in \. Their line of intersection is therefore perpendicular 
to V, or is " seen on end " in V, as the point /. Project / to W, 
where it appears as the line ;_/'. This line is the intersection of the 
two planes. 

The points ; and /, where this line of intersection ;;' meets the 
circular intersection whose center is at p, are the " intersections of 
the intersections/'" and are points on the required curve. 

N I 




Fig, 



Planes like T, at various heights on the cylinder, determine pairs 
of points on the curve of intersection on H. From H and V the 
points may be plotted on S by the usual rules of projection, thus 
completing the solution. 

68. Singular or Critical Points. — It is nearly always found that 
one or two points on the line of intersection may be projected di- 
rectly from some one view to the others without new construction 
lines. In this case a and c in \, Fig. 57, may be projected at once 



74 



Engiis^eering Descriptive Geometry 



to H and S- They correspond theoretically to points determined 
by a central plane, cutting H in a trace PP\ h and d may also be 
projected directly, as they correspond to planes whose traces in 
H are BB' and DD\ These critical points should always be the 
first points identified and recorded, though usually no explanation 
will be given, as they should be obvious to any one who has grasped 
the general method. 

QB. A Cone Intersected by an Inclined Plane. — Fig. 59 shows 




the descriptive drawing of a right circular cone intersected by an 
inclined plane whose traces are JK and KL. Two methods of solu- 
tion are shown. 

A plane R, containing the axis PP', and therefore perpendicular 
to fi, is shown by its traces R'R and RR". It intersects the cone 
in the elements Pj and PJc. From fi project these points / and Jc 
to V, and draw the elements in V- The V projection of the inter- 
section of R with the plane JK is the line JK, and the points e and 
f are the intersections of the intersections, e and / are now pro- 



Intersections of Curved Surfaces 



75 



jected to the plan, where they necessarily lie on the line ER". Sym- 
metrical points e' and f are also plotted and all four points trans- 
ferred to the side elevation. 

A plane T perpendicular to the axis PP' whose traces are T'T 
and TT" may be used instead of Pi. Its intersection with the cone 
is a circle, seen on edge in the front elevation as the line hh'. Its 
center is g, and radius is gli. Draw this circle in the plan. The 




Fig. 60. 



intersection of T with the plane JK is a line, seen on end, as the 
point / of the front elevation. Draw /'/ in W as this line. The 
points / and /' are the intersections of the intersections. 

70. Intersection of Two Cylinders. — Fig. 60 shows the inter- 
section of two cylinders. Since they are right cylinders, and their 
axes are at right angles, planes parallel to any one of the three 
reference planes will cut only straight or circular elements of the 
cylinders. By the solution. Fig. 60, auxiliary planes parallel to V 



76 



Engineering Descriptive Geometry 



liave been cliosen, the traces of one being WB and BE". This plane 
intersects the vertical cylinder in the lines 1:1:' and IV, and it inter- 
sects the horizontal cylinder in the lines mm' and nn'. The inter- 
sections of these intersections are the points marked r. 

If the axes of the cylinders do not meet but pass at right angles, 
no new complication is introduced. If the axes of the cylinders 
meet at an angle, and one or both cylinders are inclined, the choice 




V '^Hwl 



of methods may be greatly reduced, but one method is always pos- 
sible. To discover it, try planes parallel to the axes of both cylin- 
ders, or parallel to one axis and to one plane of reference; or in 
some manner bearing a definite relation to the nature of the sur- 
faces. 

71. Intersection of a Cylinder and a Sphere. — In Fig. 61 a 
sphere is intersected by a cylinder, whose axis PP' does not pass 
through the center of the sphere at ^. In the solution. Fig. 61, 



Intersections of Curved Surfaces 



77 



auxiliary planes parallel to V liave been chosen, the traces of one of 
them being B'R and RR". The plane R cuts the sphere in a circle 
whose diameter is eg, as given by the plan. This circle is described 
in V- The intersections of this circle with the elements of the 
cylinder Tck' and IV are the points marked r, points on the required 
curve of intersection. 

In this case the points are first determined on the front elevation 
and then projected to the side elevation. Solutions by planes par- 
allel to H or to S iiiay be made, requiring however different con- 
struction lines. 




72. Intersection of a Cone and a Cylinder: Axes Intersecting. — 

In Fig. 62 a cone and a cylinder intersect at right angles. The 
solution chosen is by horizontal planes, as T. 

An alternate solution is by planes perpendicular to S> and con- 
taining the point P. The planes must cut both surfaces, and their 
traces, where seen on edge, as PR, Fig. 62, must cut the projections 
of both surfaces. These two solutions hold good even if the axes 
do not meet but pass each other at right angles. 



78 



Engineering Descriptive Geometry 



If the axes are not at right angles, modifications must be made, 
and the search for a system of planes making elementary intersec- 
tions with both surfaces requires some ingenuity and thought. 

73. Intersection of a Cone and Cylinder: Axes Parallel. — A 
simple case is shown in Fig. 63. Two methods of solution are avail- 
able. In one, horizontal j)lanes are used. Each plane, such as Tj 




makes circular intersections, with both cone and cylinder, the inter- 
sections intersecting at points t and t. A second method is by 
planes perpendicular to ff, containing the axis PP'. One plane 
'' B " is shown by its traces P'P in Yi and PP" in S» this plane 
being taken so as to give the same point t on the curve and another 
point t' . In the execution of drawings of this class it is natural to 
take the auxiliary planes at regular intervals if the planes are 
parallel to each other, or at equal angles if tlie planes radiate from 
a central axis. 



Intersections of Curved Surfaces 79 

Problems VII. 

• 70. An inclined cylinder has one base in H and one in a plane 
parallel to H- Its axis is P (11, 8, 0), P' (5, 8, 16) . Its radius is 
4 units. It is intersected by a plane perpendicular to \, whose 
trace passes through the points (5, 0, 0) and (11, 0, 16). Draw the 
three projections and show one intersecting auxiliary plane by con- 
struction lines. 

71. A cone has its vertex in Hi at (6,6,0) and its base in a 
plane parallel to f\, center at (6, 6, 12), and radius 5. It is inter- 
sected by a plane containing the axis of Y and making angles of 
45° with W and S- Draw the projections. 

72. A cone has its vertex at (2, 14, 16) and its base is a circle 
in W, center at (8, 8, 0), and radius 6. Find its intersection with 
a vertical plane 4 units from g. 

73. A right circular cylinder has its base in S, center at (0, 8, 8), 
and radius 4. Its axis is 16 units long. Another right cylinder 
has its base in H, center at (8, 8, 0), radius 5, and axis 16 units 
long. Draw their lines of intersection, the smaller cylinder being 
supposed to pierce the larger. 

74. A right circular cylinder has its base in S, center at (0, 7, 8), 
and radius 4. Its axis is 16 units long. Another right circular 
cylinder has its base in J-f, center at (8, 9, 0), radius 5, and axis 16 
units long. Draw their line of intersection, the smaller cylinder 
being supposed to pierce the larger. 

75. Two inclined circular cylinders of 3 units radius have their 
bases in H and in f\' (16 units from H), The axis of one is 
P (4, 8, 0), P' (12,8,16), and of the other is Q (12,8,0), 
Q' (4,8,16). Prove that their intersection consists of two parts, 
one a circle in a plane parallel to f\., and one an ellipse in a plane 
parallel to S- 

76. A sphere has its center at (8, 9,8), and radius 6-| units. A 
vertical right circular C3rlinder has its top base in f\, center at 
(8,6,0), radius 4, and length 16 units. Find the intersection of 
the surfaces. 

77. A right circular cylinder, axis P (0,8,9), P' (16,8,9), 
radius 5, is pierced by a right circular cone. The base of the cone 



80 Engineering Descriptive Geometry 

is in a plane 16 units from ff, center at Q' (8, 8, 16), and radius 
6. The vertex of the cone is at ^ (8, 8, 0) . Find the lines of inter- 
section. 

78. An inclined cjdinder has an oblique line P (0,11,5), 
P' (16, 5, 11) for its axis. The radius of the circular base is 4 
units and the planes of the bases are S^ and S' parallel to S at 16 
units' distance. The cylinder is cut by a plane parallel to V at 7 
units' distance from V- Draw the three projections of the cylinder 
and the line of intersection. 

79. An inclined cylinder has an oblique line P (0,11,5), 
P' (16, 5, 11) for its axis. The radius of the circular base is 4 
units, and the planes of the bases are S^ and S' parallel to S at 16 
units' distance. The cylinder is cut by a plane perpendicular to 
V, its trace passing through the points (2,0,0) and (14,0,16). 
Draw the three projections. 



CHAPTER VIII. 

INTERSECTIONS OF CURVED SURFACES; CONTINUED. 

74. Intersection of a Surface of Revolution and an Inclined 
Plane. — In Figs. 64 and 65 a surface of revolution is shown. It is 




Fig. 65. 



cut by an inclined plane perpendicular to H in the first case, and 
by one perpendicular to V in the second case. The planes are 
given by their traces, and the problem is to find the curves of inter- 
section. Both solutions make use of cutting planes perpendicular 
to PP', the axis of revolution of the curved surface. 



82 



Engineeeing Descriptive Geometry 



In Fig. 64 a plane T, taken at will perpendicular to PP', cuts 
the surface of revolution in a circular element seen as the straight 
line at' in V« ^ is projected to W and the circle aW drawn. The 
inclined plane whose traces are JK and KL is intersected by the 
plane T in a line whose horizontal projection is the line JK itself. 
t and t' (on W) a^"© therefore the intersections of the intersections 
and are projected to the front elevation, giving points on the re- 
quired line of intersection. A system of planes such as T defines 
points enough to fully determine the curve, mtt'n. 

In Fig. 65 the given plane has the traces IX and XZ. The plane 
T intersects the surface of revolution on the circle atcf, and it 




L_T 



intersects the plane in the line W, seen on end in V as the point t. 
t and f in ff are points on the required curve of intersection, mtt'n. 

The point of this surface of revolution APC has been given a 
special name. It is an " ogival point." The generating line AP 
is an arc of 60°, center at C, and conversely the generating line PO 
has its center at A. The shell used in ordnance is usually a long 
cylinder with an ogival point. A double ogival surface is produced 
by revolving an arc of 120° about its chord. 

75. Intersection of Two Surfaces of Revolution: Axes Par- 
allel. — This problem is illustrated in Fig. 66, where two surfaces of 



Inteksections of Curved Surfaces 



83 



revolution are shown. A horizontal plane T cuts both surfaces in 
circular elements. These elements are drawn in fi as circles abed 
and efgh. t and f are the intersections of the intersections. From 
H t and f are projected to V and S. The problem in Art. 73 is 
but a special case of this general problem. In addition to the solu- 
tion by horizontal planes another solution is there possible, due to 
special properties of the cone and cylinder. 






67. 

76. Intersection of Two Surfaces of Revolution: Axes In- 
tersecting. — An example of two surfaces of revolution whose axes 
intersect is given by Fig. 67. A surface is formed by the revolution 
of the curve ww' about the vertical axis PP', and another surface 
by revolving the curve uQ about the horizontal axis QQ'. The in- 



84 Engineeeing Descriptive Geometry 

tersection of the axes PP' and QQ' is the point p. The peculiarity 
of this case is that no plane can cut both surfaces in circular ele- 
ments. However, a sphere described with the point of intersection 
of the axes as a center, if of proper size, will intersect both surfaces 
in circular elements. V is parallel to both axes and on this pro- 
jection a circle is described with p as center representing a sphere. 
The radius is chosen at will. To keep the drawing clear, this 
sphere lias not been described on plan or side elevation, as it would 
be quite superfluous in those views. 

The sphere has the peculiarity that it is a surface of revolution, 
using any diameter as an axis. The curve wiv' and the semicircle 
mab?i are in the same plane with the axis PP'. When both axes 
are revolved about PP', a and h, their points of intersection, gene- 
rate circular elements, which are common to the sphere and to the 
vertical surface of revolution. Therefore, these circles are the in- 
tersections of the sphere and the vertical surface. The H and S 
projections of these circles are next drawn. 

The curve uQ and the semicircle qcdr are in the same plane with 
the axis QQ'. When both axes are revolved about QQ', their inter- 
sections, c and d, generate circles which are common to both sur- 
faces, or are their lines of intersection. The circle generated by c 
is drawn in H and S? ^ut that generated by d is not needed. 

The three circles aa', bb', and cc' appear as straight lines on V> 
but from them the points t and s, the intersections of the intersec- 
tions, are determined. These are points on the required curve in 

V. 

The circle aa' appears as a circle ata't' in ff, and as a line if 
in S- The circle cc' appears as a circle ctc'f in g, and as a line ee' 
in H- These circles intersect in H at ^ and f, and in S at ;^ and f 
and s and s'. These are points on the required curves in H and g. 

For the complete solution, a number of auxiliary spheres, differ- 
ing slightly in radius, must be used. 

77. Intersection of a Cone and a Non-Circular Cylinder. — A 
non-circular cAdinder is a surface created by a line which moves 
always parallel to itself, being guided by a curve lying in a plane 
perpendicular to the generating line. This curve, called the direc- 
trix, is usually a closed curve. The cross-section of such a cylinder 
is everywhere similar to the directrix. 



Inteesections of Curved Surfaces 



85 



This fact may be utilized to advantage in some cases. In Fig. 
G8, an oblique cone and a non-circular cylinder intersect. The 
directrix of the cylinder is a pointed oval curve, ahcd in ff. Hori- 
zontal planes, as T'T, intersect the cylinder in a curve identical in 
shape with its directrix, so that its projection on H coincides with 
the projection of the directrix on ff. The intersection with the 
cone is a circle, mt'tn, and the intersections of the intersections are 
the points t. 




78. Alteration of a Curve of Intersection by a Fillet. — In Fig. 
69 a hollow cone and a non-circular cylinder, ahcd in H? intersect. 
On the left half the unmodified curve of intersection is traced by 
the method of the preceding article, no construction lines being 
shown however, as the case is very simple. On the right half the 
curve is modified by a fillet or small arc of a circle which fills in 
the angular groove. The fillet whose center is at g modifies that 
point of the line of intersection marked c. The top of the circular 
arc marks the point where an H or S projector is tangent to the 
surface. 
7 



86 



Engineering Descriptive Geometry 



The corresponding crest to the fillet at other positions on the 
curve of intersection is traced as follows : If a line drawn through 
Z; and parallel to PG, the generating line of the cone, is used as 
a new generator it will by its rotation about PP' create a new 
cone, on the surface of which the required line of the crests of the 




Fig. 69. 



fillets must lie. ,If a line mn, parallel to cc', the generating line of 
the cylinder, is moved parallel to cc', and at a constant distance 
from the surface of the non-circular cylinder, it will generate a 
new non-circular cylinder on the surface of which the required 
path of the point k must lie. The directrix of this new cylinder is 
drawn in ff, the line rms, as shown. The intersection of these two 



Intersections of Curved Surfaces 



87 




Fio. 70. 



88 Engineering Descriptive Geometry 

new surfaces, found by the method used above (or by planes per- 
pendicular to H through the axis PP'), is the required path of Tc 
or the line which appears on V ^iid S- The line rms, representing 
the same path on ff, is not properly a line of the drawing and is 
not inked except as a construction line. 

79. Intersection of a Helicoidal Surface and a Plane. — In Fig. 
70 there is shown a long-pitched screw having a triple thread, such 
as is often employed for a " worm." To the left is shown a partial 
longitudinal section giving the generating lines. In V the con- 
cealed parts of the helical edges are omitted, except in the cases of 
one of the smaller and one of the larger edges. The plane whose 
trace on V is KL is perpendicular to the axis, and terminates the 
screw threads. The intersection of this plane with the screw 
threads is the curve of intersection to be drawn on H- It is deter- 
mined by passing planes containing the axis of the worm. One of 
these is shown by its traces PR and RR'. 

From points a and h in the plan corresponding points are plotted 
on the front elevation, a falling on the helix of small diameter 
(extended in this case), and h on the helix of large diameter. This 
element ah of the helix is seen to pierce the plane KL at Jc. This 
point Jc is projected to the plan and is one of the points on the 
required curve mTcn. 

Problems VIII. 

(For units, use inches on blackboard or wire-mesh cage, or small 
squares on cross-section paper.) 

80. An anchor-ring is formed by revolving a circle of 6 units 
diameter about a vertical axis, so that its center moves in a circle 
of 10 units diameter, center at ^ (8,8,8). The anchor-ring is 
intersected by a plane parallel to V passed through the point 
A (8, 6, 8) and by another plane parallel to V through the point 
B (8,4,8). Draw the projections of the ring, the traces of the 
planes and the lines of intersection. 

81. The same anchor-ring is intersected by a plane perpendicu- 
lar to V; having a trace passing through the points C (0, 0, 2) and 
D (8, 0, 8). Make the descriptive drawing and show the true shape 
of the lines of intersection. 



Intersections of Curved Surfaces 89 

82. The same anchor-ring is intersected hy a right circular 
cylinder, axis P (12,8,0), P' (12,8,16), and diameter 4 units. 
Make the descriptive drawing of the anchor-ring, imagining it to 
be pierced by the cylinder. 

83. An anchor-ring has an axis P (0,8,8), P' (16,8,8). Its 
center moves in a plane 7 units from S? describing a circle of 8 
units diameter. The radius of the describing circle is 3 units. It 
is intersected by an ogival point whose axis is a vertical line 
Q (7, 8, 3|), Q' (7, 8, 16). The generating line of the ogival point 
is an are of 60°, with center at (0,8,16), and radius 14 units. 
The point Q is the vertex and the point Q' is the center of the circu- 
lar base of 7 units radius. The axes intersect at p (7, 8, 8). Draw 
the projections and the line of intersection, front and side eleva- 
tions only. 

84. The line P (4, 13, 8), P' (16, 8, 8) is the chord of an arc of 
90°, Avhose radius is 9.2 units. The arc is the generating line of a 
surface of revolution of which PP' is the axis. Draw the projection 
on f\. Draw the end view on an auxiliary plane HJ perpendicular 
to PP', the trace of U on H intersecting OX at (16, 0, 0). The 
surface is intersected by a plane perpendicular to ]HI and contain- 
ing the line PP'. Draw the line of intersection on V- 

85. The same surface is intersected by a plane perpendicular to 
H whose trace in f\ passes through the points (4, 16, 0) and 
(16,5,0). Draw the line of intersection on V- 

86. The line P (3, 8, 8), P' (13, 8, 8) is the chord of an arc of 
90°, radius 7.07 units. It is the axis of revolution of a surface of 
which the arc is the generating line. It is intersected by a right 
circular cone having its vertex at ^ (8, 8, 2), and center of base at 
Q' (8, 8, 12), radius of base 5 units. Draw the line of intersection. 

87. A non-circular cylinder has its straight elements, length 16 
units, perpendicular to W- The directrix is a smooth curve 
through the points .4 (14, 6, 0), B (12, 4, 0), C (10, 4, 0), 
D (8, 5, 0), P (5, 8, 0), P (2, 13, 0). It is pierced by a cylinder 
whose base is in \, whose axis is perpendicular to V at the point 
(8,0,8), whose radius is 5 units, and whose length is 14 units. 
Find the line of intersection in S. 



90 Engineering Descriptive Geometry 

88. The line P (8, 8, 2), P' (8, 8, 14) is the axis of a right cir- 
cular cylinder of 6 units diameter. Projecting from the cylinder is 
an helicoidal surface, of 12 units pitch, of which G (5,8,2), 
G' (1, 8, 2) is the generating line. The helicoid is intersected 
by a plane perpendicular to Ji whose trace in H passes through the 
points (5,0,0) and (16,11,0). Draw the plan and front eleva-' 
tion of the cylinder and helicoid and plot the line of intersection 
with the plane. 

89. The helicoidal surface of Problem 87 is intersected by a right 
circular cylinder whose axis ^ (12, 8, 2), Q' (12, 8, 14) is parallel 
to PP'. The radius of the cylinder is 3 units. Draw the line of 
intersection. 



CHAPTER IX. 
DEVELOPMENT OF CURVED SURFACES. 

80. Meaning of Development as Applied to Curved Surfaces.— 

Many curved surfaces may be developed on a plane in a manner 
smiilar to the development of prisms and pyramids explained in 
Articles 45 and 46. By development, is meant flattening out, 
without stretching or otherwise distorting the surface. If a curved 
surface is developed on a plane and this portion of the plane, called 
" the development of the surface/' is cut out, this development may 




Fig. 71. 



be bent into the shape of the surface itself. The importance of 
the process comes from the fact that many articles of sheet metal 
are so made. If a sheet of paper is bent in the hands to any fan- 
tastic shape, it will always be found that through every point of 
the paper a straight line may be drawn on the surface in some one 
direction, the greatest curvature of the surface at this point being 
in a direction at right angles to this straight line element through 
the point. The surfaces which can be formed by twisting a plane 
surface without distortion are called surfaces of single curvature. 
The curved surfaces, therefore, which are capable of development 
are only those which are surfaces of single curvature and have 
straight line elements, but not by any means all of these. All forms 



92 



Engineering Descriptive Geometry 



of cylinders and cones, right circular, oblique circular, or non- 
circular, may be developed. The helicoidal surfaces, illustrated by 
Figs. 55 and 56, though having straight elements, cannot be de- 
veloped, nor can the hyperboloid of revolution, a surface generated 
by revolving a straight line about a line not parallel nor intersect- 
ing. Figs. 71 and 72 are perspective drawings showing the process 
of rolling out or developing a right circular cylinder and a right 
circular cone. 

81. Rectification of the Arc of a Circle. — In developing curved 
surfaces it frequently happens that the whole or part of tlie cir- 
cumference of a circle is rolled out into a straight line. Since the 
surface must not be stretched or compressed, the straight line must 
be equal in length to the arc of the circle. This process of finding 
a straight line equal to a given arc is called rectifying the arc. K'o 



This angle not 
, to exceed 60" 




Fig. 73. 



absolutely exact method is possible, but methods are known which 
are so nearly exact as to lead to no appreciable error. These have 
the same practical value as if geometrically perfect. 

In Fig. 73, AB is the arc of a circle, center at C. For accurate 
work the arc should not exceed 60°. It is required to find a 
straight line equal to the given arc. Draw AH, the tangent at one 
extremity, and draw AB, the chord. Bisect AB at D. Produce the 
chord and set off AE equal to AD. With ^ as a center, and with 
EB as a radius, describe the arc BF, meeting AH at F. Then 
i4F=arc AB, within one-tenth of one per cent. 

In this figure, and in the two following ones, the arc and the 
straight line equal to it are made extra heavy for emphasis. 



Development of Curved Sureaces 



9-i- 



82. Rectifying a Semicircle. — A second method, applicable par-- 
ticularly to a semicircle, was recently devised by Mr. George Pierce. 
In Fig. 74 the semicircle AFB is to be rectified. A tangent BC, 
equal in length to the radius, is drawn at one extremity. Join AC,. 
cutting the circumference at D. Lay off DE = DC, and join BE^ 
producing BE to the circumference at F. Join AF. Then the 
triangle AEF, shown lightly shaded, has its periphery equal to the 
semicircle AFB, within one twenty-thousandth part. The peri- 
phery may be conveniently spread into one line by using A and E 
as centers, and with AF and EF as radii, swinging F to the left to 
and to the right to H on the line AF extended. GH is the recti- 
fied length of the semicircle. 

83. To Lay OS an Arc Equal to a Given Straight Line. — This 
inverse problem, namely to lay ojff on a given circle an arc equal to 



f\/ A, exceed 60" 



\ 



>\ 




A D 



B 




Fig. 75. 



Fig. 76. 



a given straight line, frequently arises. In Fig. 75 a line AB is 
given. It is required to find an arc of a given radius AC equal to 
the given line AB. At A erect a perpendicular, making AC equal 
to the given radius, and with C as a center describe the arc AF. 
On AB, take the point D at one-fourth of the total distance from 
A. With B as center and DB as a radius, draw the arc BF, meet- 
ing AF at F. AF is the required arc, equal to AB. 

This process is also accurate to one-tenth of one per cent if the 
arc AF is not greater than 60°. If in the application of this process 
to a particular case the arc AF is found to be greater than 60°, the 
line AB should be divided into halves, thirds or quarters, and the 
operation applied to the part instead of to the whole line. 



94 Engineering Descriptive Geometry 

84. Development of a Straight Circular Cylinder. — In Fig. 60 
let the intersecting cylinders represent a large sheet-iron ventilat- 
ing pipe, with two smaller pipes entering it from either side. Such 
a piece is called by pipe fitters a " cross." The problem is to find 
ihe shape of a flat sheet of metal which, when rolled up into a 
cylinder, will form the surface of the vertical pipe, with the open- 
ings already cut for the entrance of the smaller pipes. Before 
developing the large cylinder, it must be considered as cut on the 
straight element BB'. After the pipe is formed from the develop- 
ment used as a pattern, the element BB' will be the location of a 
longitudinal seam. 

A rectangle. Fig. 76, is first drawn, the height BB' being equal 
to the height of the cylinder and the horizontal length being equal 
to the circumference of the base BCD A. (This length may be best 
found by Mr. Pierce's method, which gives the half-length, BD.) 
On the drawing, Fig. 60, the base BCDA must be divided into 
equal parts, 24 parts being usually taken, as they correspond to 
■arcs of IS""', which are easily and accurately constructed with the 
draftsman's triangles. Only 6 of these 24 parts are required to be 
actually marked on Fig. 60, as the figure is doubly symmetrical 
and each quadrant is similar to the others. On Fig. 76 the line 
BCDAB is divided into 24 parts also, the numbering of the lines 
of division nmning from to 6 and back to for each half-length 
of the development. In V of Fig. 60, draw the elements corre- 
sponding to the points of division. The elemnt W already drawn 
corresponds to jSTo. 4, and BB' and CC correspond to l^os. and 6. 
The others are not drawn in Fig. 60, to avoid complicating the 
figure, but would have to be drawn in practice before constructing 
the development. On the four elements which are numbered 4 on 
the development, Fig. 76, lay off the distances Ir equal to Ir in 
Fig. 60. On the two elements. Fig. 76, numbered 6, lay off Cc or 
Aa equal to Cc of Fig. 60, and imagine the proper distances to be 
laid off on elements numbered 3 and 5. Smooth curves through 
the points thus plotted are the ovals which must be cut out of the 
sheet of metal to give the proper-shaped openings for the small 
pipes. 

When it is known in advance that the surface of such a cylinder 



Development of Curved Surfaces 95 

as that in Fig. 60 must be developed, it is often possible to so 
choose the system of auxiliary intersecting planes used to define 
the curve of intersection as to give the required equally spaced 
straight elements for the development. 

The smaller cylinder may be developed in the same way. A new- 
system of equally spaced straight elements would probably have to 
be chosen for this C3flinder. 

85. Development of a Right Circular Cone. — The cone of Fig. 
63 has been selected for this illustration. Imagine it to be cut on 
the element PB and flattened into a plane. The surface takes the 




Fig. 77. 



form of a sector of a circle, the radius of the sector being the slant 
height of the cone (or length of the straight element), and the arc 
of the sector being equal in length to the circumference of the base 
of the cone. Several means of finding the length of the arc of the 
sector are available. 

The most natural method is to rectify the circumference of the 
base and then, with the slant height as radius, to draw an arc and 
to lay out on the arc a length equal to this rectified circumference. 
In Fig. 63 suppose that the semi-circumference ABC (in W) has 
been rectified by Pierce's method. In Fig. 77 let an are be drawn 
with radius PB equal to PB in S? Fig. 63, and from B draw a 
tangent BE equal to one-half the rectified length of the semi-cir- 
cumference. Find the arc BC equal to BE by the method of Art. 



96 Engineeeixg Descriptive Geometry 

83, Fig. 75. BC is one-fourth of the required arc, and corresponds 
to the quadrant BC in f\, Fig. 63. Divide the arc BC and the 
quadrant BC into the same number of equal parts, numbering 
them from to 6, if 6 parts are chosen. Eepeat the divisions in 
the arc CD (equal to BC), numbering the points of division from 
6 down to 0, this duplication of numbers being due to the symmetry 
of the M projection of Fig. 63, about the line AFC. In Fig. 63, 
as in Fig. 77, the points to 6 are all supposed to be joined to P, 
the only straight elements actually shown there being PO, P4, and 
P&. 

On the elements P4 of the development lay off the true length 
of the line Pt (and the true length of the line Pf also). Pt is an 
oblique line, but if its ff projector-plane (Pt in ff, Fig. 63) be 
revolved up to the position Pm, the point i in V moves to m, and 
Pm is the true length of Pt. The distance Pg (Y, in Fig. 63) is 
laid off on P6 of the development. 

When the proper distances have been laid off on the elements 
P2j P3 and P5, a smooth curve may be drawn through the points. 
The sector, with this opening cut in it, is the pattern for forming 
the cone out of sheet iron or any thin material. 

If the ratio of PA to P'A in V, Fig. 63, can be exactly deter- 
mined, the most accurate method of getting the angle of the sector is 
by calculation, for the degrees of arc in the development are to the 
degrees in the base of the cone (360°) as the radius of the base of 
the cone is to the slant height. In this case P'A is f P.4. The 
sector in Fig. 75 subtends f x360°, or 216°. In the use of this 
method a good protractor is required to lay out the arc. 

Problems IX. 

90. Draw an arc of 60° with 10 units radius. At one end draw 
a tangent and on the tangent lay off a length equal to the given 
arc. On the taligent lay off a length of 8 units, and find the length 
of arc equal to this distance. 

91. An arc of 12 units radius, one of 9 units radius, and a 
straight line are all tangent at the same point. Find on the tan- 
gent the straight line equal in length to 45° of the large arc. Find 
the length on the other arc equal to this length on the tangent and 
show that it is an arc of 60°. 



Development of Curved Surfaces 97 

92. Eectify a semicircle of 10 units radius and compare this 
length with the calculated length, 31.4 units. 

93. A rectangle 31.4 units by 12 units is the developed area of a 
cylinder of 10 units diameter. A diagonal line is drawn on the 
development, which is then rolled into cylindrical form. Plot the 
form taken by the diagonal and show that it is a helix of 12 units 
pitch. 

94. A right circular cone has a base of 10 units diameter, and 
a vertical height of 12 units. Its slant height is 13 units. Calcu- 
late the angle of the sector which is the developed surface of iihe 
cone. Find this angle by rectifying the circumference of the base 
of the cone, and by finding the arc equal to the rectified length. 
(This last operation must be performed on one-third or one-quarter 
of the rectified length, to keep the accuracy within one-tenth of 
one per cent.) 

95. A semicircle, radius 10 units, is rolled up into a cone. What 
is the radius of the base? What is the slant height? What is the 
relation between the area of the curved surface of the cone and the 
area of the base ? 

96. A right circular cylinder, such as Fig. 49, is of 7.59 units 
diameter, and 12 units height. It is intersected by a plane per- 
pendicular to V through the points C and A'. Draw plan, front 
elevation and the development of the surface. 

97. A right circular cone, like that of Fig. 51, has its front ele- 
vation an equilateral triangle, each side being 10 units in length. 
From .4u a perpendicular is drawn to PvGv cutting it at E. If this 
line represents a plane perpendicular to V? draw the development 
of the cone with the line of intersection of the cone and plane traced 
on the development. 

98. A right circular cjdinder, standing in a vertical position, as 
in Fig. 49, diameter 7 units, and length 10 units, is pierced from 
side to side by a square hole 3^ units on each edge, the axis of the 
hole and the axis of the cylinder bisecting each other at right 
angles. Draw the development of the surface. 

99. A sheet of metal 22 units square with a hole 11 units square 
cut out of its middle, the sides of the hole being parallel to the 
edges of the sheet, is rolled up into a cylinder. Draw the plan, 
front and side elevations of the cvlinder. 



CHAPTER X. 

STRAIGHT LINES OF UNLIMITED LENGTH AND THEIR 

TRACES. 

86. Negative Coordinates. — We have dealt only with points hav- 
ing positive or zero coordinates, and the lines and planes have been 




Fig. 78. 



limited in their extent, or, if infinite, have extended indefinitely 
only in the positive directions. As it becomes necessary at times 
to trace lines and planes in their course, no matter if they cross 
the reference planes into new regions of space, the use and meaning 
of negative coordinates must be explained. The value of the x 
coordinate of a point is the length of the S projector or perpen- 
dicular distance from the point to the side reference plane S. (See 
Figs. 6 and 7, Art. 9.) If this value decreases gradually to zero. 



Lines of Unlimited Length: Their Traces 



99 



the point moves towards S until it lies in g itself. If this value 
becomes negative, it is clear that the point crosses the side reference 
plane into a space to the right of it. 

For example, a point P, having a variable x coordinate, but hav- 
ing its y coordinate always equal to 4 and its z coordinate equal to 
2, is a point moving on a line parallel to the axis of X. If x de- 
creases to zero, it is on S at the point marked Pg in Fig. 78. If 
the X coordinate decreases further, reaching a value of —3, it 
moves to the point P in that figure. Fig. 78 is the perspective 
drawing of a point P ( — 3, 4, 2 ) . The y and z projectors cannot 
project the point P to H and V in their customary positions, but 



H ••E,ext.t\ 



X" 



V 



;t 



k 



e| %-% X 



and ^v 

Vejctended. 










^7} 



P., 



ivp- 



9 P 



...^\ 



H 










N 



fs Y, 



s 



Fig. 79. 



Fig. 80. 



Z\ 
Fig. 81. 



project it upon parts of those planes extended beyond the axes of 
Y and Z, as shown. In Fig. 79, the corresponding descriptive 
drawing, it must be understood that the plane H? extended, has 
been revolved with H, about the axis of X, into the plane of the 
paper, V> and S has been revolved as usual about the axis of Z, 
coming into coincidence with V? extended. This "development" 
of the planes of reference is exactly as described in Art. 7. It is 
noticeable that the x coordinate of P is laid off to the right of the 
origin instead of to the left. Ph lies, therefore, in the quadrant 
which usually represents no plane of projection, and Pi, lies in the 
quadrant which usually represents S- Ps lies in its customary- 
place, since both y and z, the coordinates which alone appear in S> 
are positive. 



100 Engineeri^tg Descriptive Geometry 

It is evident that the laws of projection for ff, V and S> Art. 
11, have not been altered, but simply extended. Pn and Pv are in 
the same vertical line; P,, and Ps are in the same horizontal line; 
and the construction which connects Ph and Pg still holds good. 

In Fig. 79 the space marked S represents not only S but V 
extended as well. 

In Fig. 80 is represented a point P (3, —2, 3), having a negative 
y coordinate. The point is in front of V? at 3 units' distance, not 
behind V- The projection on ff, instead of being above the axia 
of X a distancee of 3 units, is below it by the same amount. So also 
the projection on S is to the left of the axis of Z, a distance of 3 
units, instead of the the right of it. After developing the reference 
planes in the manner of Art. 7, plane ff, extended, has come into 
coincidence with V? and plane S? extended, has also come into co- 
incidence with V- Thus the field representing V represents also 
the other two reference planes, extended. 

In Fig. 81 a point P (3,3,-3) having a negative z coordinate 
is represented. The point is above JHI 3 units, instead of below H^ 
at the same perpendicular distance. P projects upon V on V 
extended above the axis of X. After developing the reference 
planes, plane H comes into coincidence with V extended. Pg is 
on ^ extended above the axis of Y, and therefore after develop- 
ment it occupies the so-called " construction space." 

Points having two or three negative coordinates may be dealt 
with in the same manner, but are little likely to arise in practice. 

It is evident that subscripts must be used invariably, to prevent 
confusion whenever negative values are encountered. 

87. Graphical Connection Between Ph and Pg. — In Figs. 79, 80 
and 81, Pn and Pg are connected by a construction line PhfhfsPs in 
a manner Avhich is an extension of that shown by Fig. 7, Art. 9. 
ISTote that the quadrant of a circle connecting Pn and Pg must be 
described always on the construction space or on the field devoted 
to V? never on the fields devoted to W or S- 

88. Traces of a Line of Unlimited length, Parallel to an Axis. — 
A straight line which has no limit to its length, but extends in- 
definitely in either direction, must necessarily have some points 
whose coordinates are negative. In passing from positive to nega- 



Lines of Unlimited Length: Their Traces 



101 



tive regions the line mtist pass through some plane of reference 
(having one of its coordinates zero at that point), and the point 
where it pierces a plane of reference is called the trace of the line 
on that plane of reference, the word trace. being used to indicate a 
" track " or print showing the passage of the line. 

Lines parallel to the axes have been used freely alread3^ An W 
projector is simply a vertical line or line parallel to the axis of Z. 
Any perspective figure showing a point P and its horizontal pro- 
jection Pft will serve as an illustration of this line, as PPn in Fig. 
C, Art. 9. 






A_ 


_\ 


\ 




Bk 




\ 


X 


Av 





AY3 




B 


z 


-Ds 



Fig. 82. 



Fig. 83. 



Imagine PPn to be extended in both directions as an unlimited 
straight line. Then P/, is the trace of the line on H. In Fig. 7, 
the point Pn itself is the H projection of the line. P^e, extended 
in both directions, is the vertical projection and Psfs is the side 
projection. Thus it is seen that a vertical line has but one trace, 
that on the plane to which it is perpendicular. PPv may be taken 
as an illustration of a line parallel to the axis of Y, and PPs of one 
parallel to the axis of X. A better example of this latter case is 
shown in Figs. 15 and 16, Art. 16. The line BAAs, perpendicular 
to S, has its trace on S at Ag. 



102 



Engineering Descriptive Geometry 



89. Traces of an Inclined Straight Line. — An inclined line snch 
as AB in Figs. 83 and 83 pierces two reference planes as at A and 
B, but as it is parallel to the third reference plane, S? it has no 
trace on §. The pecnliarity of the descriptive drawing of this line, 
Fig. 83, is the apparent coincidence of the H and V projections 
as one vertical line. The S projection is required to determine the 
traces A and B. 

90. Traces of an Oblique Straight Line : The ff and V Traces. — 
An oblique line, if unlimited in length, must pierce each of the 
reference planes, since it is oblique to all three. x\ny line is com- 




FiG. 84. 



Fig. 85. 



Fig. 86. 



Fig. 87. 



pletely defined When two points on the line are given. If two 
traces of a straight line are given, the third trace cannot be assumed, 
but must be constructed from the given conditions by geometrical 
process. It will always be found that of the three traces of an 
oblique line one trace at least has some negative coordinate. 

As the complete relation between the three traces is somewhat 
complicated, the relation between two traces, as, for instance, ff 
and V traces, must be considered first. Two cases are shown, the 
first by Figs. 84 and 85, and the second by Figs. 86 and 87. The 
line AB is the line whose traces are A (5,0,4) and B (3,4,0). 
The line CD is the line whose traces are C (7, 0, 5) and D (3, 4, 0). 



Lines of Unlimited Length: Their Traces 



103 



From the descriptive drawing of AB, Fig. 85, it is seen that the 
li projection of the line cuts the axis of Z vertically above the 
trace on Y, and that the V projection cuts the axis of Z vertically 
under the trace on H- It may be noted that the two right triangles 
A],BBy and BvAAh have the line AhB^ on the axis of Z as their 
common base. From the descriptive drawing of the line CD, Fig. 
87, it is seen that the effect of the vertical trace C having a nega- 
tive z coordinate simply puts C (on V) above Cn, instead of below it. 
The two right triangles ChDDv and DvCCn have the line CkDv on 
the axis of X, as their common base, but the latter triangle is above 
the axis instead of in its normal position. 




91. Traces of an Oblique Straight Line: The V and S Traces. — 

Figs. 88 and 89 show two lines piercing V ai^d S- 

The line AB pierces \ 2it A and S at B. The two right triangles 
AsABv and BvBAg have their common base AgBv on the axis of Z. 
The line CD pierces V at C and S extended at D, the point D 
having a negative y coordinate. The right triangles CsCDv and 
DvDCs have their base DvCs in common on the axis of Z, but in the 
descriptive drawing D^DCs lies to the left of the axis of Z instead 
of to the right, owing to the point D having a negative y coordinate. 



104 



Engikeering Descriptive Geometry 



92. Traces of an Oblique Straight Line : The ff and S Traces. — 

Figs. 90 and 91 show two lines piercing ff and S. 

The line AB pierces H at A and g at B. The triangles AgABh 
and BhBAs have their common base AgBn on the axis of Y, Fig. 90, 
but in the descriptive drawing the duplication of the axis of Y 
causes this base AgBh to separate into two separate bases, one on 
OYji and one on OYg. Otherwise, there has been no change. 

The line CD pierces JHI at C and S extended at D, the point B 
having a negative z coordinate. In Fig. 90 CsCDn and DnDCs have 




their common base CgDi, on the axis of Y, but in the descriptive 
drawing CsDj, appears in two places. The triangle DnDCs lies above 
S in the " construction space," or on S extended, since D has a 
negative z coordinate. 

93. Three Traces of an Oblique Straight Line.— Figs. 92 and 93 
show an oblique straight line ABC piercing V at A, ff at B, and 
S extended at C. Since the line is straight, the three projections 
of the line AB^Gv, AsBgC and A-hBCh are all straight lines. In the 
perspective drawing. Fig. 92, part of the V projection is on V ex- 
tended and part of the S projection on g extended. 



Lines of Unlimited Length: Their Traces 105 

In the descriptive drawing, Fig. 93, the relation between A and 
B is the same as that in Fig. 85, as shown by the two triangles 
AhABv and BvBAj,, or the quadrilateral AnAB^B. The relation 
between A and C, as shown by the quadrilateral AsACvC, is the 
same as that between A and B, Fig. 89, as shown by the quadri- 
lateral AsABvB. The relation between B and C, Fig. 93, as shown 
by the two triangles BgBCh and CiiCBg, is the same as that between 
C and D of Fig. 91, as shown by the triangles CsCDh and DhDCg. 
No new feature has been introduced. 



Snxall part 
ojV extended 




Fig. 92 



94. Paper Box Diagram. — To assist in understanding Figs. 92 
and 93, a model in space should be made and studied from all 
sides. The complete relation of the traces is then quickly grasped. 
Construct the descriptive drawing. Fig. 93, on coordinate paper, 
using, as coordinates for A, B and C, (15,0,12), (5,12,0), and 
(0, 18, —6). Fold into a paper box after the manner of Fig. 9, 
Art. 12, having first cut the paper on some such line as mn, so that 
the part of the paper on which C is plotted may remain upright, 
serving as an extension to S- It will be found that a straight wire 
or long needle or a thread may be run through the points A, B and 
C, thus producing a model of the line and all its projections. 



106 Engineering Descriptive Geometry 

95. Intersecting Lines. — If two lines intersect, their point of 
intersection, when projected upon any plane of reference, must 
necessarily be the point of intersection of the projections on that 
plane. For example, a line AB intersects a line CD at E. Project 
E upon a plane of reference, as ff. Then En must be the point of 
intersection of AjiBn and CnDh. In the same way Ev must be the 
point of intersection of AvBv and CvDv, and Es of AgBs and CsDs. 

To determine whether two lines given by their projections meet 
in space or pass without meeting, the projections on at least two 
reference planes must be extended (if necessary) till they meet. 
Then for the lines themselves to intersect, the points of intersec- 
tion of the two pairs of projections must obey the rules of pro- 
jection of a point in space (Art. 11). Thus if AjtBh and ChDh are 
given and meet at a point vertically above the point of intersection 
of AvBv and CvDv, the two lines really meet at a point whose pro- 
jections are the intersections of the given projections. If this con- 
dition is not filled the lines pass without meeting, the intersecting 
of the projections being deceptive. 

96. Parallel Lines. — If two lines are parallel, the projections of 
the lines on a reference plane are also parallel (or coincident). 
For, the two lines make the same angle with the plane of pro- 
jection; their projector-planes are parallel; and the projections 
themselves are parallel. 

Thus if a line AB is parallel to another line CD, then AnBh must 
be parallel to ChDh, AvBv to CvD^, and AgBs to CsDs. If the two 
lines lie in a plane perpendicular to a plane of projection — for 
example, perpendicular to ff — then the ff projector-planes coin- 
cide and the ff projections also coincide. The V and g projections 
are parallel but not coincident. 

If two lines do not fill the conditions of intersecting or of parallel 
lines, they must necessarily be lines which pass at an angle without 
meeting. 



Lines of Unlimited Length: Theih Teaces 107 

Problems X. 

100. Plot the points A (8, 6, -4), B (7, -3,5), C (-7, 0, 12). 

101. Plot the points 4 (6, -10, 3), 5 (0,0,-8), (7 (-6,5,4). 

102. Make a descriptive drawing of a line 26 units long from 
the point P ( — 8, 4, 9), perpendicular to g. What traces does it 
have ? What are the coordinates of its middle point ? 

103. A line is drawn from P (12,5,16) perpendicular to Hi. 
Make the descriptive drawing of the line, and of a line perpen- 
dicular to it, drawn from Q (0, 0, 8). What is the length of this 
perpendicular line, and where are its traces? 

104. A straight line extends from A (8,12,0) through D 
(8, 6, 8) for a distance of 20 units. Make the descriptive drawing 
of the line. Where are its traces and its middle point ? 

105. A straight line pierces H at A (8, 6, 0) and V at 5 
(8, 0, 12). Draw its projections. Where is its trace on S? What 
are the coordinates of D, its middle point? 

106. A straight line extends from E (15,6,16) through A 
(3, 6, 0) to meet S- Make the descriptive drawing and mark the 
traces on H and S- 

107. Draw the lines A (16, 11, 8), B (4, 8, 2) ; C (12, 5, 10), 
D (0,2,4); and E (11,3,0), F (5,15,8). Which pair meet, 
which are parallel, and which pass at an angle? What are the 
coordinates of the point of intersection of the pair which meet ? 

108. The points A (8, 0, 12), B (0, 8, 6) and (-8, 16, 0) are 
the traces of a straight line. Make the descriptive drawing of the 
line. 

109. The points A (8, -4,0), D (4,4,6) and E (2,8,9) are 
on a straight line. Find the trace B where it pierces V and the 
trace C where it pierces S- 



CHAPTER XI. 
TLANES OF UNLIMITED EXTENT: THEIR TRACES. 

97. Traces of Horizontal and Vertical Planes. — The lines of 
intersection of a plane with the reference planes are called its 
traces. Planes of unlimited extent may be of three kinds, parallel 
to a reference plane, inclined, or obliqne. Unlimited planes of the 
first two classes have been dealt with already, but for the sake of 
precision may be treated here again to advantage. 

A horizontal plane is one parallel to H, and the trace of such a 
plane on V is a line parallel to the axis of X, and the trace on S 
is a line parallel to the axis of I'. These traces meet the axis of Z 
at the same point and appear on the descriptive drawing as one 
continuous line. There is of course no trace on H- In Fig. 58, 
Art. 67, the plane T, represented by its traces T'T on V and TT" 
on S. is a horizontal plane. These traces are not only the intersec- 
tions of T with ff and S- ^'^^t T is " seen on edge '' in those views. 
Every point of the plane T, when projected upon V? lies somewhere 
on the line T'T, extended indefinitely in either direction. 

A vertical plane parallel to V has for its traces a line on JHI 
parallel to the axis of X, and on S a line parallel to the axis of 
Z, with no trace on V- These traces ineet the axis of Y at the 
same point, and appear on the descriptive drawing as two lines at 
right angles to this axis, the point on Y separating into two points 
as usual. In Fig, 57, Art. 66, a vertical plane B, parallel to Y, is 
represented by its traces E'E on ff and EE" on g. 

A vertical plane parallel to S has for its trace on H a line paral- 
lel to the axis of Y, and for its trace on V a line parallel to the 
axis of Z, with no trace on S- These traces meet the axis of X at 
the same point and appear on the descriptive drawing as one con- 
tinuous line. 

98. Traces of Inclined Planes. — Inclined planes are those per- 
pendicular to one reference plane, but not to two reference planes. 
The auxiliary planes of projection have been of this kind. In 



Plaxes of Unlimited Extent: Their Traces 



109 



Fig. 20, Art. 22, the plane U, perpendicular to H, has the line 
MX for its trace on ff, and XN for its trace on V- In the de- 
scriptive drawing. Fig. 21, 31 X and XNv are these traces. 

If in Fig. 20 both U and § are imagined to be extended towards 
the eye, they will intersect in a line parallel to OZ. This g trace 
will be on S extended, and every point of it will have the same 
negative y coordinate. Of the three traces of U, two are vertical 
lines, and one only, MX, is an inclined line. The plane in Fig. 64, 
Art. 74, may be taken as a second example of an inclined plane 
perpendicular to H. The trace on S is not a negative line in this 
case, but is a vertical line on § to the right of the axis of Z at a 
distance equal to OJ. 

In Fig. 57, Art. 66, IJ, JK and KL are the three traces of an 
inclined plane perpendicular to V. In every case of an inclined 
plane the inclined trace is on that reference plane to which it is 
perpendicular, and shows the angles of the inclined plane with, one 
or both of the other reference planes. 




99. Traces of an Oblique Plane: All Traces " Positive."— The 

general case of an oblique plane is shown in Fig. 94. The plane 
P is represented as cutting the cube of reference planes in the lines 
marked PH, PV and PS. These lines are the traces of the plane 
P, and may be understood to extend indefinitely, the plane itself 
extending in all directions without limit. They are shown limited 
in Fig. 94 in order to make a more realistic appearance. PH, PV 
and PS are used to define the three traces. 



110 



Engineekixg Descrtptite Geometry 



Where FE and PV meet y^e have a point common to three phmes, 
P. H and V- Since it is common to H and V it is on the line of 
intersection of ff and V- or in o^li"?r words it is on the axis of X. 
This point is marked a. In the same way PH and PS meet at h 
on the axis of Y, and PI' and PS meet at c on the axis of Z. 

The descriptive drawing. Fig. 95, is obvious from the explana- 
tion of the perspective drawing. From Fig. 95 it is evident that 
if two traces of a plane are given the third trace can be determined 




Fig. 96. 



Fig. 97. 



by geometrical construction. Thus, if PH and PV are given, PS 
may be defined by extending PH to h on the axis of Y and extend- 
ing PV to c on the axis of Z. The line joining he is the required 
trace of the plane on S- If any two points on one trace are given, 
and any one point on a second trace, the whole figure may be com- 
pleted. Thus any two points on PH define that line and enable a 
and h to be found. A third point on PT^ taken in conjunction 
with a, defines PV, and enables c to be located, he, as before, 
defines the trace PS. This is an application of the general prin- 
ciple tliat three points determine a plane. 



Planes of Unlimited Extent: Their Traces 111 

100. Traces of an Oblique Plane: One Trace "Negative." — In 

Figs. 94 and 95 the plane P has been so selected that all traces have 
positive positions. These are the portions "usually drawn. Of 
course each trace may be extended in either direction, points on 
the trace then having one or more negative coordinates. Any 
trace having points all of whose coordinates are positive, or zero 
may be called a positive trace. 

In Fig. 96 a plane P is shown, intersecting H and Y in the 
" positive " traces PH and PV. The third trace, PS, in this case, 
has no point all of whose coordinates are positive. In the descrip- 
tive drawing, Fig. 97, the two positive traces, meeting at a on the 




Fig. 98. 



Fig. 99. 



axis of X, are usually considered as fully representing the plane P. 
From these lines PH and PV, alone, the imagination is relied upon 
to " see the plane P in space,'' as shown by Fig. 96. 

In Fig. 98, the plane Q is represented. Ordinarily the positive 
traces QV and Q8, meeting at c on the axis of Z, are the only 
traces shown in the descriptive drawdng. Fig. 99, and are considered 
to indicate perfectly the path of the plane Q. 

101. Position of the Negative Trace. — The negative trace P8, 
in Fig. 96, is shown as one of the edges of the rectangular plate 
representing the unlimited plane P. This line PS has been de- 
termined by extending PH to meet the axis of Y (extended) at 



112 Engineerixg Desckiptive Geometry 

h, and by extending PV to meet the axis of Z (extended) at c. 
The line joining h and c is the trace PS. It will be noted that in 
finding the location of PS in Fig. 97, PV has been extended to cut 
the axis of Z (extended np from ZO) at c and PH has been ex- 
tended to cut the axis of Y (extended down from YO) at h. 1) 
has been rotated 90° about the origin, and the points h and c thus 
plotted (on S extended) have been found to give the line PS. 
Ever}^ step of the process and the lettering of the figure have been 
similar to those used in finding PS from PR and PV in Art. 98. 

In Fig. 98, the negative trace is QH, the top line of the rect- 
angular plate representing the unlimited plane Q. QH has been 
determined as follows: QV extended meets the axis of X extended 
at a, and QS extended meets the axis of Y extended at h. The line 
ah is therefore the trace on ff, or QH. In the descriptive drawing 
the same process of extending QV to a and ^^S' to & determines the 
line QH, a line every point of which has some negative coordinate. 
Of course QH must be considered as drawn on parts of the plane 
fi extended over V? S? and the so-called construction space. In 
finding the negative traces, it is imperative to letter the diagrams 
uniformly, keeping a for the intersection of the plane with the axis 
of X, h for that with the axis of Y, and c for that with the axis of 
Z. With this rule & will always be the point which is doubled by 
the separation of the axis of Y into two lines, and the arc hh will 
always be described in the construction space or in the quadrant 
devoted to V? never in those devoted to H and S- 

102. Parallel Planes. — If two planes are parallel to each other, 
their traces on ff, V and S are parallel each to each. This prop- 
osition may be proved as follows : If we consider two planes P 
and Q parallel to each other and each intersecting the plane H, the 
lines of intersection with ff (PH and QH) cannot meet, for, if 
they did meet, the planes themselves would meet and could not then 
be parallel planes. PH and QH must therefore be parallel lines 
described on ff . Thus, if a plane P and a plane Q are parallel, 
then PH and QH are parallel, PV and QV are parallel, and PS 
and QS are parallel. 

The method of finding the true length of a line by its projection 
upon a plane parallel to itself, treated in Chapter III, is really the 



Plaxes of Unlimited Extent: Their Traces 113 

process of passing a plane parallel to a projector-plane of the given 
line. Thus in Fig. 21, Art. 22, the auxiliary plane HJ has its hori- 
zontal trace X2I parallel to AnBj,, and' the vertical trace of the H 
projector-plane, if drawn, vs'ould be parallel to XN^. 

103. The Plane Containing a Given Line. — If a line lies on a 
plane, the trace of the line on any plane of reference (the point 
where it pierces the plane of reference) must lie on the trace of the 
plane on that plane of reference. Thus, if the line EF, Fig. 100, 
lies on the plane P, then A, the trace of EF on fil, lies on PE, the 
trace of P on H ; and B, the trace of EF on V, lies on PV, the 
trace of P on V« 




Fig. 100. 



Fig. 101. 



From this fact it follows that to pass a plane which will contain 
a given line it is necessary to find two traces of the line and to pass 
a trace of the plane through each trace of the line. As an infinite 
number of planes may be passed through a given line, it is neces- 
sary to have some second condition to define a single plane. For 
example, the plane may be made also to pass through a given point 
or to be perpendicular to a reference plane. 

In Fig. 100, if only the line EF is given and it is required to pass 
a plane P, containing that line, and containing also some point, 
as a, on the axis of X, the process is as follows : Extend the line 
EF to A and B, its traces on H and V- Join Ba and aA. These 



114 Engineering Descriptive Geometry 

are the traces of the required plane P. In the descriptive drawing, 
Fig, 101, the corresponding operation is performed, A and B 
must be determined as in Art. 90, and joined to a. These lines 
represent the traces of a plane containing the line EF and the 
chosen point a. 

To pass a plane Q containing the line EF and also perpendicular 
to H (Figs. 100 and 101), the trace of (J on H must coincide with 
the projection of EF on ff, for the required plane perpendicular to 
lil is the IHI projector-plane of the line. Its traces are therefore 
ABn and BnB. 

The traces of a plane containing EF and perpendicular to V are 
BAv and AvA. 

104. The Line or Point on a Given Plane. — To determine whether 
a line lies on a given plane is a problem the reverse of that just 
treated. It amounts simply to determining whether the traces of 
the line lie on the traces on the plane. Thus, in Fig. 101, if PV 
and PH are given, and the line EF is given by its projections, the 
traces of EF must be found, and if they lie on PH and PV the line 
is then knoMoi to lie on the given plane P. 

To determine w'hether a given point lies on a given plane is 
almost as simple. Join one projection of the point with any point 
on the corresponding trace of the plane. Find the other trace of 
the line so formed, and see whether it lies on the other trace of the 
given plane. Thus in Fig. 101, if the traces PH and PV and the 
projections of any one point, as E, are given, select some point on 
PH, as .4, and join E^A and EvAv. Find the trace B. If it lies 
on PVj, the point E itself lies on P. 

To draw on a given plane a line subject to some other condition, 
such as parallel to some plane of reference, is always a problem in 
constructing a line whose traces are on the traces of the given 
plane, and which yet obeys the second condition, whatever it may be. 

105. The Plane Containing Two Given Lines. — From the last 
article, if a plane contains two given lines, the traces of the plane 
must contain the traces of the lines themselves. The given lines 
must be intersecting or parallel lines, or the solution is impossible. 

In Fig. 102 two lines, AB and AC, are given by their projections. 
They intersect at A, since An, the intersection of the ff projections, 



Planes of Unlimited Extent: Their Traces 115 

is yertically above Av, the intersection of the V projections. Ex- 
tend the lines to E, F, G and H, their traces on ffi and V- Join 
the IM traces, E and G, and produce the line also to a on the axis 
of X. Join the V traces, H and F, and extend the line HF also 
to a. Ea and aH are thg traces of a plane P containing both lines, 
AB and AC. The meeting of the two traces at a is a test of the 
accuracy of the drawing. 

This process may be applied to a pair of parallel lines, but not of 
course to two lines which pass at an angle without meeting. 




Fig. 102. 



106. The Line of Intersection of Two Planes. — If two planes 
P and Q are given by their traces, their line of intersection must 
pass through the point where the lil traces meet and the point 
where the V traces meet. Thus, in Fig. 103, PH and QH meet 
at A and PV and QV meet at 5. A and B are points on the 
required line of intersection of P and Q, and since A is on H and 
B is on V, they are the H and V traces of the line of intersection. 
AB}, and BAy are therefore the projections, and should be marked 
PQn and PQ,. 



116 



Engineering Descriptive Geometry 




X a 



Fig. 103. 




Planes of Unlimited Extent: Their Traces 



117 



107. Special Case of the Intersection of Two Planes : Two Traces 
Parallel. — The construction must be varied a little in the special 
case when two of the traces of the planes are parallel. In Fig. 104 
the traces PV and QV are parallel. In carrying out the construc- 
tion as in Fig. 100, it is necessary to join Av with B. But the 
point B is the intersection of PV and QV, which are parallel, and 
is therefore a point at an infinite distance in the direction of those 
lines, as indicated by the bracket on Fig. 104. To join Ay with B 
at infinity means to draw a line through Av parallel to PV and QV. 





PH ^^ 


\ 




OH 


X 




~^N \ 


^% 


■\^ \ 


X 


PQv 





J^ ^^ 




:^% 




PV 




A S 




QV \^ 


z 





Fig. 105. 



From B, at infinity, a perpendicular must be supposed to be drawn 
to the axis of X, intersecting it at Bh. Bu is therefore at an infinite 
distance to the right on the axis of X (extended). To join the 
point A with the point Bh means, therefore, to draw a line through 
A parallel to the axis of X. These lines are the required projec- 
tions of PQ. 

108. Special Case of the Intersection of Two Planes: Four 

Traces Parallel. — Another special case arises when the four traces 

(on two planes of projection) are parallel. It is then necessary to 

refer to a third plane of projection. In Fig. 105 the planes P and 

9 



118 



Engineering Descriptive Geometry 



Q have their four traces on Hi and V all parallel. The planes axe 
inclined planes perpendicular to S, a^d i^ their traces are drawn 
on S, their intersection is the line PQ. In S both P and Q are 
" seen on edge/' so their line of intersection is " seen on end.'' 
From PQs, PQv and PQi, are dra'O'n by projection. 

109. The Point of Intersection of a Line and a Plane. — The 
simple cases of this problem have been previously explained and 
used. If the plane is horizontal, vertical or inclined, there ia 




Fig. 106. 



always one view at least in which it is seen on edge. In that view 
the given line is seen to pierce the given plane at a definite point 
from which, by the rules of projection, the other views of the point 
of intersection are easily determined. Thus in Fig. 27, Art. 38, 
the point a, where PA pierces the plane KL, is determined first in 
V and then projected to H and S- 

The general case of tbis problem may be solved as in Fig. 106. 
A plane P is given by its traces PE and PI'. A line AB is- given 
by its projections. It is required to find where AB pierces P. The 



Planes of Unlimited Extent: Their Traces 119 

solution is as follows: Let a plane perpendicular to V be passed 
through the projection AvBr. According to Art. 103 the traces of 
this plane are BvFv and FvF. Draw the line of intersection of this 
plane with the plane P (Art. 106) as follows: B^.F^ and PV in- 
tersect at E. F and E are the traces of the line of intersection of 
the two planes. Complete the drawing of the line of intersection 
in n, as FEn. 

Eeferring to the horizontal projection, AnBh is seen to intersect 
FEh, the H projection of the line of intersection, at W,,. Since 
both FE and AB are lines which lie in the vertical projector-plane 
through AB, this point of intersection, Wj,, is the projection of the 
true point of intersection, W, of those two lines. From Wn project 
to Wv for the other projection of IF. This point W which lies on 
P and is on the line AB is the required point. 

Problems XI. 

(For blackboard or cross-section paper or wire-mesh cage.) 

110. Plot the point A (4,7,9). Pass a horizontal plane P 
through the point A, and draw the traces of P. Also a vertical 
plane Q, parallel to V^ and draw its traces. Also an inclined plane 
R, perpendicular to H? making an angle of 45° with OX. 

111. Plot the line A (8,2,4), B (2,6,16). Pass an inclined 
plane P perpendicular to ff through this line and draw the traces 
of P. At C, the middle point of AB, pass a plane Q perpendicular 
to P and to U, and draw QH and QV. 

112. The plane P cuts the axes at the points a (10,0,0), 
h (0, 5, 0) and c (0, 0, 15). Pass a plane Q parallel to P, through 
the point a' (6,0,0). 

113. A plane P has its trace on H through the points 
A (12,12,0) and & (0,6,0). Its trace on V passes through the 
point c (0, 0, 12). Draw the three traces. Draw three traces of a 
plane Q, parallel to P through the point (f (3, 0, 0). 

114. An indefinite line contains the points A (11, 2, 6) and 
B (5,6,0). Pass a plane P perpendicular to ff containing this 
line and -draw the traces PH, PV and PS. Pass a plane Q con- 
taining this line and the point a' (2,0,0). Draw the traces QH 
and QV. Draw the negative trace QS on S extended over |HI. 



120 Engineering Descriptive Geometry 

115. A plane P cuts the axis of X at a (4, 0, 0), the axis of Y 
at ?J (0, 6, 0), and the axis of ^ at c (0, 0,-12). Draw its traces. 
Draw the V a^d S traces of a plane Q parallel to P and containing 
the line A (1, 4, 11), B (4, 1, 14). 

116. An inclined plane, perpendicular to H, has for its V and 
S traces lines parallel to OZ at positive distances of 15 and 5 units. 
An inclined plane Q perpendicular to fi has its V and S traces 
parallel to OZ at distances of 12 units and 8 units. Draw all three 
traces and the projections of PQ, their line of intersection. 

117. Draw the traces of a plane P, containing the points 
A (8, 1, 3), B (4, 5, 1) and C (2, 4, 3). Does the point D (4, 1, 5) 
lie on this plane? 

118. The traces of a plane P are lines through the points 
a (10,0,0), l (0,15,0) andJ5' (14,0,6). A plane Q has its 
traces through the points a' (2,0,0), E, and F (7,5,0), Draw 
the projections of their line of intersection, PQ. 

119. The plane P cuts the axes at a (12, 0, 0), & (0, 12, 0) and 
c (0, 0, 12). Where does the line K (1, 5, 12), L (5, 3, 6) pierce 
the plane? 



CHAPTER XII. 

VARIOUS APPLICATIONS'. 

110. Traces of an Inclined Plane Perpendicular to an Oblique 
Plane. — One of the most general devices used in the drafting room 
is the auxiliary plane of projection, and it is often advantageous 
to pass this plane perpendicular to some plane of the drawing in 




Fig. 108. Fig. 109. 



order to get the advantage of showing that plane " on edge." Thus 
in Fig. 31, Art. 43, the plane \] has been taken perpendicular to 
the long rectangular faces of the triangular prism, in order to 
show clearly where BB' and DD' pierce those planes. The manner 
of passing the plane U was fairly clear in that case from the 
simplicity of the figure. However, as it is not always clear how to 
pass a plane perpendicular to an oblique plane, the general method 
may well be explained here. In Fig. 107 the plane P, previously 
shown in Fig. 94, is represented, and an auxiliary plane \], per- 
pendicular to it and to H, is shovra. The traces of P are PR, PV 
and PS as before, and the traces of \J are UH and US. It must 



122 Engineeking Descriptive G-eometrt 

be understood that the ff traces of these planes, PH and UE, are 
perpendicular to each oilier, as this condition is essential if P and 
\] are to be planes perpendicular to each other. 

Fig. 108 is the descriptive drawing corresponding to the per- 
spective drawing. Fig. 107. At some point It on PH a line Mdh 
has been drawn perpendicular to PH. This line is the inclined 
trace of a plane HJ perpendicular to ff. The other traces of U are 
parallel to the axis of Z (Art. 98). One of these, the trace on S, 
is shown by the line dgNg, parallel to OZ, dn and dg being two 
representations of the same point d in Fig. 107, just as hn and hg 
represent the point &, duplicated. Mdn may be called UH and dgNa 
may be called US. UH and US are the traces of an inclined plane 
\], perpendicular to the oblique plane P. 

The proof that P and U are perpendicular to each other is as 
follows: If, in Fig. 107, a line hli' is drawn perpendicular to W 
at the point li, it will lie in the plane HJ. The angle ahh' will 
then be an angle of 90°, and by construction the angle ahd is also 
90°. Thvis the line all is perpendicular to two intersecting lines de- 
scribed in the plane \] and is therefore perpendicular to U itself. 
The plane P contains the line PH and is thus perpendicular to \]. 

111. An Auxiliary Plane of Projection Perpendicular to an 
Oblique Plane. — ^To utilize the inclined plane \] as an auxiliary 
plane of projection, its developed position must be shown by drawing 
diiNu perpendicular to UH. This line is the duplicate position of 
dgNg or US. In developing the planes, U is first revolved on UH 
as an axis into the plane of H as shown in Fig. 109, and then with 
H into the plane of the paper, V- The trace of P on \], or PU, is 
the line of intersection of the planes, and is shown clearly in Fig. 
107. This line passes through li where PH and UH meet, and 
through s where PS and US meet. In Fig. 108, dns is laid off on 
djiNu, eq^ial to d^s, and the line lis is the required trace of P on U, 
or PU. The actual line PU, in Fig. 108, is only that part of PU, 
in Fig. 107, which is between li and s, shown as a broken line. 

The important part in this process is that U is taken perpen- 
dicular to P, so that P is " seen on edge " on \]. By this process 
the plane P, which is oblique when ff, Y and S are considered. 



Various Applications 



123 



becomes an inclined plane when only ff and \] are considered 
As it is easier to deal with inclined than with oblique planes, 



we 




Fig. 110. 



may now treat P as inclined to H and perpendicular to U in 
further operations. 

Fig. 108 is well adapted to making a paper k)x diagram which. 



124 Engineeeixg Descriptive Geometry 

when folded, will give most of the lines of Fig. 107. To reconstruct 
Fig. 108, plot the points a (18,0,0), h (0,18,0), c (0,12,0), 
d (0,6,0), h (6,12,0) and s (0,6,8). The line dnNu is at an 
angle of 45° with ZOYu and the construction space YsOd],Nu can 
be folded away inside by creasing or cutting it on several lines. 

112. Intersection of an Oblique Plane and a Cylinder. — An ex- 
ample of the use of an auxiliary view on which an oblique plane is 
seen on edge is shown in Fig. 110, An inclined cylinder is inter- 
sected by an oblique plane P given by its traces PH, PV and PS. 
It is required to describe on the cylinder the curve of intersection 
of the plane and the cylinder. The solution is as follows : An 
auxiliary plane U, perpendicular to P and to ff, is chosen, and 
PU is drawn upon HJ as in Fig. 108. PC is the view of P " seen on 
edge" in \]. Auxiliary cutting planes parallel to ff are used for 
the determination of the required line of intersection. The traces 
of one of the planes are drawn, as TT in V, TT" in S, and T"T"' 
in U- This latter trace is parallel to di,M (or UH), because T is 
parallel to M, and the distance dnT" is equal to dsT" in S- T"T"' 
cuts the axis of the cylinder at p. p is projected to ff, and the 
circular element described in ff, with p as a center, is the inter- 
section of the auxiliary plane T and the cylinder. In \] the planes 
P and T are both " seen on edge," intersecting in a line " seen on 
end." This point projected to H gives this line of intersection of 
P and T as W. 

The intersections of the intersections are therefore the points t 
and f, where the circle and the straight line meet. 

113. The Angle between Two Oblique Lines. — This problem of 
finding the angle between two oblique lines is shown in Fig. 111. 
Let two lines AB and AC, meeting at A, be given by their ff and 
V projections. It is required to find the true angle between them. 

By the process of Art. 105, Fig. 102, the traces of the plane con- 
taining AB and AC are found and the lines are all lettered accord- 
ing to Fig. 102. 

An auxiliary plane of projection, \], is passed perpendicular to 
PV, and therefore perpendicular to both P and V, and is revolved 
into the plane V- The projections of AB and AC on this plane 



Various Applications 



125 



fall in the single line AuCuBu, since P, the plane of the lines, is 
" seen on edge " on U- ^ portion of the plane P is now revolved 
about the U projector of the point A into a position parallel to 
XM. In U> C'w moves to C"„ and Bu to B'u, revolving about A as 
their center. In Y> Bv moves to B'v and Cv to C'v, both parallel to 
XM. This is the process of finding the true length of a line by 
revolving about a projector, as in Art, 33. AvB'v is the true length 




Fig. 111. 



of AB; AvC'v is the true length of AC; and B'vAvC'v is the true 
angle between the lines. 

This process makes it possible to find the true shape of any 
figure described on an oblique plane. 

114, A Plane Perpendicular to an Inclined Line. — It is often 
advantageous to pass a plane perpendicular to a line in order to 
use the plane as a plane of projection, on which the given line will 
be seen on end as a point. The method of passing a plane perpen- 
dicular to an inclined line is shown in Fig. 112. Let AB be an 
inclined line, lying in a plane parallel to V? so that AiiBh is parallel 



126 



Engineering Descriptive Geometry 



to the axis of X. It is required to find the traces of a plane P 
perpendicular to AB. The essential point is that the traces of the 
plane must be perpendicular to the corresponding projections of 
the line. Thus, choose some point p on the inclined projection of 
the line, in this case on AvB^, and through p draw a perpendicular 
to AvB^, to serve as the trace of P. At a, where this trace FY 









Y 




Br 






K 




£ 










X "k 




^K"" 




\ 


X 






-Bv 






Z 





Fig. 112. 



meets the axis of X, erect a perpendicular to FE. These lines FY 
and FR are the traces of an inclined plane perpendicular to AB 
and to V. It is noticeable that the inclined trace of the plane is 
on that reference plane which shows the inclined projection of the 
line.* 



_ * A proof that F is perpendicular to AB is as follows- AB 
IS the line of intersection of its own fi projector-plane, and its 
own V projector-plane. F is perpendicular to both these projector- 



Yaeiods Applications 



137 



115. Application of a Plane Perpendicular to a Line. — In Fig. 
113 an application of an inclined plane perpendicular to an in- 
clined line is made for the purpose of finding the line of intersection 
between an inclined cone and an inclined cylinder whose axes do 
not meet. 




Fig. 113. 



If from P, the vertex of the cone, a line Pp is drawn parallel to 
QQ', as shown, any plane which contains this line and cuts both 

planes. For, P is perpendicular to V a^^^ therefore to the H pro- 
jector-plane, which is parallel to Vj the V projector-plane is per- 
pendicular to v., so that it is seen on edge on V j^ist as is P itself ; 
apAv is therefore the true angle between these two planes, and by 
construction is a right angle. P is therefore perpendicular to both 
projector-planes and therefore to the line AB, which is their line 
of intersection. 



128 Engineering Descriptive Geometry. 

surfaces will cut only simple elements of the surfaces. For such 
a plane contains the vertex of the cone, and therefore, if it cuts 
the cone, will cut it in straight elements; and such a plane is 
parallel to QQ', and therefore, if it cuts the cylinder, cuts only 
straight elements. ISTo other planes can be found which cut simple 
elements and can be used to determine the line of intersection. 

If a plane U is passed perpendicular to Pp at any point p, and 
is used as an auxiliary plane of projection, Pp will be seen on end 
as the point P, and any plane R through P, seen on edge in \], as 
RR', will cut only straight elements on the two curved surfaces. 
The complete projections of the cone and cylinder have been shown 
on U, and the plane R cuts the bases at a, h, c and d. These points 
projected to V enable the elements to be dra-^vn there, and the 
intersections of the intersections are the four points marked r. 
From V these points are projected to Ji and S. Two of these 
points r have been projected to the other views to show the neces- 
sary construction lines. 

116. A Plane Perpendicular to an Oblique Line. — To pass a 
plane perpendicular to an oblique line, it is only necessary to draw 
the traces of the plane perpendicular to the corresponding pro- 
jections of the line. In Fig. 114, let AB be an oblique line. At 
any point on Aj,Bh draw a perpendicular line PH. From a, where 
PH meets the axis of X, draw PV perpendicular to AB* 

A paper box diagram traced from Fig. 114, or constructed on 
coordinate paper, using the coordinates A (10, 4, 4) and B (6, 8, 2), 
C (2, 12, 0) and D (14, 0, 6), and a (8, 0, 0), will assist materially 
in understanding the problem. 

The oblique plane P is not serviceable as an auxiliary plane of 
projection. 

117. The Application of Axes of Projection to Mechanical 
Drawings. — Descriptive Geometry is a geometrical science, the 
science dealing primarily with orthographic projectioi*, while Me- 
chanical Drawing is the art of applying these principles to the 



* The proof of this construction is more difficult than in the 
corresponding case of an inclined line, but it depends as before 
on the line AB being the intersection of its ff and V projector- 
planes, and these planes themselves being perpendicular to P. 



Various Applications 



129 



needs of engineers and mechanics in the pursuit of industries. 
Mechanical Drawing includes therefore many abbreviations and 
conventional representations, which seek to curtail unnecessary- 
work and often to conve}' information as to methods of manu- 
facture and other such commercial considerations foreign to the 
strict scientific study. 




Fig. 114. 



In Mechanical Drawing many lines necessary to the strict execu- 
tion of a descriptive drawing are omitted as unnecessary to the 
application of the principles, when once the principles have been 
fully grasped. A noteworthy omission is the axes of projection, 
which, though absent, still govern the rules for making the draw- 
ing. Instead of measuring distances from the axes for every point 



130 Engineering Descriptive Geometry 

on the drawing, the " center lines " of the different views (which 
really represent central planes) are laid off and distances from 
these center lines are thereafter used. This is the regular pro- 
cedure in drawing-room practice. That this difference is purely 
one of omission is clear from the fact that axes of projection may 
always be inserted in a mechanical drawing. If two views only of 
a piece are presented, any line between them (perpendicular to the 
lines of projection from one view to another) may be selected as 
the axis of X, and any convenient point on that line as the origin 
of coordinates. 

If three views are given, as, for example, Fig. 32, Art. 44, sup- 
posing the axes to be there omitted, a ground line XOY ^ may be 
selected at will, dividing the fields of W and V- The other line 
must be determined as follows : By the dividers take the vertical 
distance from OX to the center line mn, and lay off this distance 
horizontally to the left from the center line of the side elevation. 
The line ZOYh may then be drawn. x\ll y coordinates of points will 
now check correctly, measured parallel to the two axes of Y, if the 
original drawing itself is accurate. 

It is thus evident that in applying Descriptive Geometry to prac- 
tical mechanical drawing we may fall back upon the use of axes of 
projection whenever the lack of them is felt. 

118. Practical Application of Descriptive Geometry. — Many 
draftsmen have picked up a knowledge of Descriptive Geometry 
without direct study of the science. This is largely due to the fact 
that, till very recently, all books on Descriptive Geometry were 
based on a system of planes of projection which are analogous to 
the methods of practical drawing in use on the continent of Europe, 
but which are little used in England, and hardly at all in the United 
States of America. It will be found, however, that in American 
drafting rooms all the usual devices of draftsmen are applications, 
sometimes almost unconscious applications, of the principles covered 
in the preceding chapters. The favorite device is the application 
of an inclined auxiliary plane of projection, suitably chosen ; next 
in importance is the rotation of the object to show some true shape ; 
while other applications are used less frequently. The methods of 
determining lines of intersection of planes and curved surfaces are 
exactly those described in Chapters IV, VII and VIII. 



Various Applications 131 

Problems XII. 

(For use on blackboard, with, cross-section paper or wire-mesh. 

cage.) 

120. The plane P has its traces through the points a (14, 0, 0), 
& (0,14,0) and c (0,0,7). Pass a plane Q, perpendicular to P 
and to jj-J, through the point A (5, 7, 0). If ^ is to be used as an 
auxiliary plane of projection, draw the trace of P on ^ when Q has 
been revolved into coincidence with If-J. 

121. Draw the traces of a plane P cutting the axes at the points 
a (12, 0,0), h (0,8,0) and c (0,0,12). Draw the traces of an 
auxiliary plane, \}, perpendicular to PH at the point A (3, 6, 0). 
Is the point B (6, 1, 4|) on the plane P? 

122. The H trace of a plane P passes through the points 
^ (12, 5, 0) and P (6, 2, 0) . Its V trace passes through C (9, 0, 6) . 
Pass an inclined plane perpendicular to W and perpendicular to 
P, through the point D (5, 9, 7). 

123. Of a plane P, PR, the horizontal trace, passes through the 
points A (5,3,0) and B (13,9,0), and PV passes through 
C (12, 0, 11). Complete the traces of P and draw the traces of a 
plane perpendicular to PV at the point D (9, 0, 8). Prove that the 
line P (9, 6, 1), P (7, 3, 2) lies on the plane P. 

124. A sjDhere of radius 7 units has its center at (7 (8, 8, 8). A 
plane P cuts the axes of projection at a (26, 0, 0), l (0, 13, 0) and 
c (0,0,13). Pass an auxiliary plane of projection \}, perpen- 
dicular to W and to P, cutting the axis of X at ^ (16, 0, 0) . Draw 
the trace of P on U- The circle of intersection of the sphere and 
the plane P is seen on edge, in U- Show the elliptical projection 
of this circle, on W, by passing auxiliary cutting planes parallel 
to U. (If this problem is solved by use of wire-mesh cage, the 
point a is inaccessible, but PR passes through E (16,5,0), and 
Py through P (16,0,5). The plane S' can be turned to serve 

asU.) 

125. Find the true shape of the triangle A (3,2,6), B (9, 6, 2), 
C (8, 0,4). Find the traces of two of the sides of the triangle and 
pass the plane \] perpendicular to the plane of the triangle and 
perpendicular to ff, and through the point D (0,7,0). 



132 Engineering Desceiptive Geometet 

126. Find the true shape of the triangle A (8,6,1), B (4, 2, 9), 
C (10, 2, 3). Find the traces of two of the sides of the triangle 
and pass the plane \] perpendicular to the plane of the triangle 
and perpendicular to Jil, and through the point D (0, 1, 0). 

127. Draw the traces of a plane P perpendicular to V ^^^ ^o 
the line A (2, 6, 9), B (8, 6, 5) at C (11, 6, 3). If this plane is 
used as an auxiliary plane of projection, what is the projection of 
A^onit? 

128. Draw the traces of a plane P perpendicular to fil and to 
the line A (3, 9, 6), B (13,4, 6), at C (17, 2, 6), a point on AB, 
(If wire-mesh cage is used for the solution, turn S' to serve as U 
and draw on it the view of AuBu.) 

129. Draw the three traces of a plane P perpendicular to the 
oblique line A (8, 12, 5), 5 (14, 3, 7). Show that all three traces 
are perpendicular to the corresponding projections of AB. 



CHAPTER XIII. 
THE ELEMENTS OF ISOMETRIC SKETCHING. 

119. Isometric Projection. — There is one special branch, of 
Orthographic Projection which is of peculiar value for represent- 
ing forms which consist wholly or mainly of plane faces at right 
angles to each other. Ordinary orthographic views are projec- 
tions upon planes parallel to the principal plane faces of the object, 
as shown in Pig. 2, Art. 4. If, however, instead of the regular 
planes of projection, the object is projected upon a new plane of 
projection, making the same angle ivitli each of the regular planes, 
an entirely different result is obtained, called an "isometric pro- 
jection" This view has the useful property that it has all the air 
of a perspective and may, witli certain restrictions, be used alone 
without other views as a full representation of the object. 

In Art. 21 the method of converting the perspective drawings 
of this treatise into isometric sketches was explained in a rough 
and unscientific way. In this chapter there is explained the method 
of making isometric sketches from models, as a step to making 
orthographic drawings or isometric drawings. 

120. Isometric Sketches of Rectangular Objects. — Figs. 19 and 
I9a are the isometric drawings of a cube. Since the line of sight 
from the eye to the point makes equal angles with H, V and S; 
the three planes must subtend the same angle at 0. XOY, YOZ 
and XOZ are each 120°. though representing angles of 90° on the 
cube. Since opposite edges of ff are parallel, it follows that each 
face of the cube is a rhombus and that the cube appears as a regular 
hexagon, all edges appearing of exactly the same length. This 
fact is the basis of the name " isometric," meaning " equal- 
measured." 

Figs. 115, IIG and 117 are sketches of other objects, all of whose 
corners are right angles. The angles at these comers appear there- 
10 



134 



Engixeeking Desckiptive Geometry 



fore like those of the cube, either as 60° or 120° on the isometric 
sketch. 

In making the isometric sketch from a model having rectangular 
faces, the first step is to put the object approximately in the iso- 




Brick 
Fig. 115. 



Half Joint 

Fig. 116. 



Mortise 5- Tenon Joint, 

Fig. 117. 




Position for 
Orthographic 
Projection ., 




Fig. 118. 



Turned 4-5 
about a veri'i- 
cai dxis. 

Fig. 119. 



Til-hed 35°-^' 
about an hori- 
zontal QKIS. 

Fig. 120. 



metric position. At any projecting corner imagine a line to project 
from the corner so as to make equal angles with the three edo-es 
■which meet at the given corner. View the object by sighting along 
this imaginary line and begin the sketch from that view. 



The Elements of Isometric Sketching 135 

If there is any difficulty in finding this line of vision directly, 
the object may be turned horizontally through an angle of 45° and 
tilted down through an angle of 35° 44'. This operation is the 
basis of the method of finding the " isometric projection." 

Figs. 118, 119 and 120 show the steps in passing from the ortho- 
graphic position to the isometric position, the model used being a 
rectangular block with a lengthwise groove cut in one face. 

121. Isometric Axes. — It will be noticed in the previous iso- 
metric figures that all lines are drawn in one of three general direc- 
tions. One of these directions is usually taken as vertical and the 
other two directions make angles of 120° with the vertical. These 
three directions are known as the isometric axes. In this sense 
the word axis means a direction, not a line. 

In plotting points from a selected origin, the x coordinates are 
plotted up and to the left, the y coordinates up and to the right, 
and the z coordinates vertically downward, as in Fig. 19a. 

122. Isometric Paper. — Paper ruled in the direction of the iso- 
metric axes is called isometric paper, and is of great assistance in 
making isometric sketches. The lines divide the paper into small 
equilateral triangles. 

In sketching, the sides of these equilateral triangles are taken to 
represent unit distances, exactly or at least approximately. Thus, 
if the model shown in Fig. 120 is a block 3" x 3" x 8'', with a 2" x 1" 
groove lengthwise along one face, some point a on the paper is 
selected, and from it distances are taken along the isometric axes, 
so that each unit' space represents one inch. 

From a three units are co^mted vertically downward, eight up, 
and to the right, and one unit, folloY/ed by a gap in the line of one 
unit, and then a second unit, up to the left. Thus all lines of the 
sketch folloAv the ruled lines as long as the dimensions of the model 
are in even inches. 

An isometric sketch made in this manner, particularly if spaces 
have been exactty counted off according to the dimensions of the 
piece, is practically an isometric drawing. If fully dimensioned, a 
sketch on plain paper proportioned by the eye is nearly as good as 
one in which spaces are counted exactly. Such sketches serve all 



136 



Engineering Descriptive Geometry 



purposes, though of course more difficult to make than those on 
isometric paper. 




Fig. 121. 



Fig. 122. 



123. Non-Isometric Lines in Isometric Sketching. — Objects 
which have a few faces and edges oblique to the principal plane 
faces may still be shown by isometric sketching. In such cases it is 
always well to circumscribe a set of rectangular planes about the 




Fig. 123 



oblique parts of the object to aid the imagination. Dimension 
extension lines should be used for this purpose. In using isometric 
paper this squaring up is done by the lines of the paper. 



The Elements of Isometric Sketching 



137 



Figs. 122, 123 and 124 are good examples of oblique lines and 
faces. Figs. 123 and 124 show also the circumscribed isometric 
lines which " square up " the oblique parts. 

124. Angles in Isometric Sketching-. — In isometric sketchino- 
angles do not, as a rule, appear of their true magnitude. Thus the 
90° angles on the faces of the brick appear in Fig. 115 as 60° or 
120°, but not as 90°. In general, the lengths of oblique or inclined 
lines depend on position, and are not subject to measurement by 
scale. 

The lines which square up oblique parts are useful in giving the 
tangent of the angle of an oblique surface. Thus in Fig. 124, the 
angle a differs in reality from the angle as it appears in either place 

marked, but the tangent of a is ^ . In Fig. 123, (9 = tan-^ ^ . In 

V n 

practice angles are often given by their tangents. Thus the slope of 




Fig. 126 



Fig. 127. 



a roof is given as " one in two " or the gradient of a railroad as 
" three per cent." 

125. Cylindrical Surfaces in Isometric Sketching. — In ortho- 
graphic drawings circles appear commonly on planes parallel to the 
three planes of projection. To illustrate the position and appear- 
ance of circles in isometric drawing in the three typical cases, Fig. 
125 represents the isometric sketch of a cube, having a circle in- 
scribed in each square face. 

Each of the faces of the cube is perpendicular to the isometric 
axis given by the intersection of the other two faces. Thus the 
square ABCD is perpendicular to the edge BF. The circle ahcd. 



138 Engineeriis^g Descriptive Geometry 

inscribed in the square ABCD, appears as an ellipse, whose minor 
axis, ef, lies on the diagonal BD of the square, BD appearing as a 
continuation of the edge FB. In all three cases, then, the minor 
axis of the ellipse lies in the same direction, on the sketch, as that 
isometric axis to which the plane of the circle is in reality perpen- 
dicular. 

The major axis is necessarily perpendicular to the minor axis, 
and lies on the other diagonal of the square. 

Since the cylinder is the curved surface most used in engineering, 
the rule may be applied to cylinders as follows : The ellipse which 
represents the circular base of any cylinder must be so sketched 
that its minor axis is in line with the axis or center line of the 
cylinder. Fig. 126 is an isometric sketch of a piece composed of 
cylinders. All the ellipses are seen to follow this rule. 

In sketching cylindrical parts of objects, it is necessary to im- 
agine them squared up by the use of isometric lines and planes. 
Thus the first steps in sketching the piece of Fig. 126 are shown 
in Fig. 127. Th& circumscribing of a square about a circle in the 
object corresponds to circumscribing a rhombus about the ellipse 
in the isometric sketch. It now remains to inscribe an ellipse in 
the rhombus. This ellipse must be tangent to the rhombus at the 
middle of each side. To sketch the ellipse, as for example the small 
end in Fig. 127, draw the diagonals of the rhombus to get the 
directions of the major and minor axes, and find the middle points 
of the sides (by center lines, through the intersection of the diagon- 
als) . It is now easy to sketch the ellipse, having four points given, 
the direction of passing through those points, and the directions of 
the major and minor axes. 

126. Isometric Sketches from Orthographic Sketches. — A good 
exercise consists in making isometric sketches from orthographic 
sketches or drawings. The three coordinate directions, x, y and z, 
must be kept in mind at all times. Fig. 128, as an example, is most 
instructive. From the orthographic sketches, Fig. 128, the iso- 
metric sketch. Fig. 129, is to be made. A point a is selected to rep- 
resent a point a on the orthographic views. The line ah is an x 
dimension and is plotted up to the left ; ac is a y dimension, and is 
plotted up to the right; while ad is a, z dimension, and is plotted 



The Elements of Isometric Sketching 



139 



vertically downward. The semicircle is inscribed in a half-rhombus, 
tangent at h, e and /. 



























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> 


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V 

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> 


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x 


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1 








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> 


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Fig. 128. 



Fig. 129. 



The cross-section lines of Fig. 128 and the isometric lines of Fig. 
129 are represented as overlapping between the figures. Some iso- 
metric paper is ruled in this manner, so that it may be used for 
both purposes. 



140 



Engineering Descriptive Geometry 



Problems XIII. 

(For blackboard or isometric paper.) 

130. Make an isometric sketch of the angle piece^ Fig. 130, using 
the spaces for 1" distances. 



— 






- 




— 


— 




— 
























— 


< 


... 





- 






























































-- 


• — 


-k 


... 


-5 


- 














































\ 
















. 




\ 


















> 


V 


















S 




















\ 












V, 








s. 












1 



















- 




— 


— 




— 




Fig. 130. 



Fig. 131. 



131. Measure the tool-chest, Fig. 131, scale, f" = l foot, and 
make a bill of material, tabulating the boards used, and recording 
their sizes, giving dimensions in the order : width, thickness, length, 
thus : 

Mark. Name. Size. Number. 

A. Top of Chest. 14" x 1" x 24". 2. 

132. A parallelepiped, 9"x6"x3", has a 3" square hole from 
center to center of the largest faces, and a 2" bore-hole centrally 
from end to end. Make an isometric sketch. 

133. Let Fig. 3, Art. 5, represent a model cut from a 1.2" cube 
by removing the center, leaving the thickness of the walls 3". Let 
the angular point form a triangle whose base is 12" and altitude 8". 
Make an isometric sketch. 



The Elements or Isometric Sketching 141 

134. A cube of 10" has a 6" square hole piercing it centrally from 
one side to the other, and a 4" bore-hole piercing it centrally from 
side to side at right angles to the larger hole. Make an isometric 
sketch. 

135. A grating is made by nailing slats f"xi"xl3", spaced ^" 
apart, on three square pieces, 1^" square, 22" long, spaced 4^" apart. 
Make an isometric sketch. 

136. Make orthographic sketches of the bracket, Fig. 122. Views 
required are plan and front elevation. (On cross-section paper use 
the unit distance for the unit of the isometric paper. On black- 
board let each unit of the isometric paper be represented by a dis- 
tance of 2".) 

137. Make isometric sketches of Fig. 11, Art. 14, and Fig. 24, 
Art. 32. 

138. Make isometric sketches of Fig. 13, Art. 15, and of Fig. 82, 
Art. 89. In Fig. 82 let A be the point (9, 8, 0) and B the point 
(9,0,12). 

139. Make an isometric sketch of Fig. 71, Art. 80, the diameter 
of the cylinder being 7 units and the length 14 units. 

140. Make an isometric sketch of Fig. 92, Art. 93, using the 
coordinates given in Art. 94. 



CHAPTER XIV. 
ISOMETRIC DRAWING AS AN EXACT SYSTEM. 

127. The Isometric Projection on an Oblique Auxiliary Plane. — 

The sketches previously considered have generall}' had no exact scale. 
Those drawn on isometric paper have a certain scale according to 
the distance which one unit space of the paper actually represents. 




If the isometric projection is derived from an orthographic draw- 
ing of the usual kind by the laws of projection, the isometric projec- 
tion so formed has of course the same scale as the original drawing. 

In Fig. 132 an isometric projection of a cube is derived from the 
orthographic drawing by the use of an inclined plane of projection. 



Isometric Drawing as an Exact System 143 

\], and an oblique airxiliary plane of projection W- The aim is 
to produce the projection on a plane making the same angle with all 
three edges of the cube meeting at any one corner. This plane must 
be perpendicular to a diagonal of the cube. In Fig. 132 this di- 
agonal is the line EC, a true diagonal, passing through the center of 
the cube, not a diagonal of one face of the cube. 

The first, or inclined, auxiliary plane \] is taken parallel to the 
V projection of EC, and therefore perpendiciilar to V and making 
an angle of 45° with ff and S- The projection of EC on \] shows 
its true length. 

The second, or oblique, auxiliar}^ plane "W is taken perpendicular 
to EC. It is oblique as regards H and Y, but, as EG is a line par- 
allel to U> aiicl "W is perpendicular to EC, "W is perpendicular to 
\J. As regards V and \], "W is an inclined plane, having its in- 
clined trace MN on U, the trace on V being a line MLv, perpendic- 
ular to ZM, the trace of U on V- The construction of this second 
projection is therefore according to the usual methods. Any point, 
as F, is projected by a perpendicular line across the trace MN and 
the distance nFio is laid off equal to mFv. 

The projection on "VV is the isometric projection of the cube and 
is full-size if the j)lan and front elevation are full-size projections. 
The edges are all foreshortened, however, and measure only j-^-^ of 
their true length. 

128. The Angles of the Auxiliary Planes. — The plane U makes 
an angle of 45° with the plane H. The plane "W makes an angle 
of 35° 44' with S, or (90° -35° 44') with V- W the side of the 
cube is taken as 1, the length of the diagonal of the face of the cube 
is \/2, and the length of the diagonal of the cube is \'^. The 

first angle is that angle whose tangent is - - , or whose sine is — y^ > 

The second angle is that angle whose tangent is — = and whose sine 

^ ^ - ^ ^2 

. V2 

129. The Isometric Projection by Rotating the Object. — In Fig. 
134 is shown a method of deriving the isometric projection by turn- 
ing the object. The plan, front, and side elevations are drawn with 



144 



Enghsteeeing Descriptive Geometry 



the object turned through an angle of 45° from the natural posi- 
tion (that in which the faces of the cube are all parallel to the 
reference planes) , The side elevation shows the true length of one 
diagonal of the cube, AG. Some point on AG extended, as K, is 
taken as a pivot, and the whole object is tilted down through an 
angle of 35° 44', bringing AG into a horizontal position, A'G'. The 




new projection of the object in V is the isometric projection. This 
process of turning the object corresponds to the turning of the 
object in isometric sketching, as shown in Figs. 118, 119 and 120. 

The isometric projection of the cube has all eight edges of the 
same length, but foreshortened from the true length in the ratio of 
V3 to V2. 

Any object of a rectangular nature may be treated by either 
process to obtain the isometric projection. 



Isometric D^AAv-I^"G as as Exact System 145 

130. The Isometric Drawing. — To make a practical system of 
drawing capable of representing rectangular objects in an unmis- 
takable manner in one view, the fact that all edges are foreshort- 
ened alike is seized upon, but the disagreeable ratio of foreshorten- 
ing is obviated by ignoring foreshortening altogether. 

An isometric drawing is one constructed as follows: On three 
lines of direction, called isometric axes, making angles of 120° with 
each other, the true lengths of the edges of the object are laid off. 
These lengths, however, are only those which are mutually at right 
angles on the object. All otlier lines are altered in shape or length. 
An isometric drawing is distinct from an isometric projection, as 
it is larger in the proportion of 100 to 83 (V3: V2). The iso- 
metric drawing of a 1" cube is a hexagon measuring 1" on each 
edge. 

131. Requirement of Perpendicular Faces. — An isometric draw- 
ing, being a single view, cannot really give " depth," or tell exactly 
the relative distances of different points of the object from the eye. 
It absolutely requires that the object drawn shall have its most 
prominent faces, at least, mutually perpendicular. The mind must 
be able to assume that the object represented is of this kind, or the 
drawing will not be " read " correctly. Even on this assumption, 
in some cases isometric drawing of rectangular objects may be 
misunderstood if some projecting angle is taken as a reentrant one. 
Thus in Fig. 133 we have a drawing which might be taken as the 
pattern of inlaid paving or other flat object. If it is taken as an. 
isometric drawing and the various faces are assumed to be perpen- 
dicular to each other, it becomes the drawing of a set of cubes. 
Curiously enough, it can be taken to represent either 6 or 7 cubes, 
according as the point A is taken as a raised point or as a depressed 
one. In other words, it even requires one to know just how the 
faces are perpendicular to each other to be able to take the drawing 
in the way intended. 

This requirement of perpendicular faces limits the system of 
drawing to one class of objects, but for that class it is a very easy, 
direct, and readily understood method. Untrained mechanics can 
follow isometric drawings more easily than orthographic drawings. 



146 



Engineering Descriptive Geometry 




Isometric Drawik-g as an Exact System 147 

132. The Representation of the Circle. — In executing isometric 
drawings, the circle, projected as an ellipse, is the one drawback to 
the system. To minimize the labor, an approximate ellipse must 
be substituted for an exact one, even at the expense of displeasing 
a critical eye. The system, if used, is used for practical purposes 
where beauty must be sacrificed to speed. In Fig. 125 the rhombus 
ABCD is the typical rhombus in which the ellipse must be inscribed. 
The exact method is shown in Fig. 43, but requires too much time 
for constant use. The following draftsman's ellipse, devised to be 
exactly tangent to the rhombus at the middle point of each side, is 
reasonably accurate. From B, one extremity of the short diagonal 
of the rhombus, drop perpendiculars Bd and Be upon opposite 
sides, cutting the long diagonal at Jc and l. With 5 as a center and 
Bd as a radius, describe the arc dc. Similarly, with D as a center, 
describe the arc ha. With Tc and I as centers, and Ted as a radius, 
describe the arcs ad and ch. The resulting oval has the correct 
major axis within one-eighth of 1 per cent, and has the correct 
minor axis within 3^ per cent. 

This draftsman's ellipse is exact where required, namely, on the 
two diameters ac and dh, which are isometric axes, and it is prac- 
tically exact at the extremity of the major axis. 

133. Set of Isometric Sketches. — Fig. 135 is a set of isometric 
sketches of the details of the strap end of a small connecting-rod, 
from which to make orthographic drawings. The isometric sketch 
is much clearer than the corresponding orthographic sketch, and 
the set shows clearly how the pieces are assembled. 

The orthographic drawing of the assembled rod end is much 
easier to make than the assembled isometric drawing. It is in fact 
clearer for the mechanic than the assembled isometric drawing 
would be, for the number of lines would in that case be quite con- 
fusing. It illustrates well the fact that isometric sketches and 
drawings should be limited to fairly simple objects. 

Another noteworthy fact is that center lines, which should always 
mark s}Tnmetrical parts in orthographic drawings, should be used 
in isometric drawing only when measurements are recorded from 
tJiem. 

The sketch as given is taken directly from an examination paper 
used at the U S. Kaval Academy for a two-hour examination. On 



148 Engineering Descriptive Geometry 

account of the shortness of the period, only one orthographic view, 
the front elevation, is required, but if time were not limited, a plan 
also should be drawn. 

The following explanation of the sheet is printed on the original : 
" Explanation of Meclianism. — ^The isometric sketches represent 
the parts of the strap end of a connecting-rod for a small engine. 
In assembling, A, B, C, and D are pushed together, with the thin 
metal liners, G, filling the space between B and C. The tapered 
key, E, is driven in the ^" holes of A and D, which will be found to 
be in line, except for a displacement of -J" which prevents the key 
from being driven down flush wdth the top of the strap D. The two 
bolts, F, are inserted in their holes, nuts H screwed on, and split 
pins (which are not drawn) inserted in the ^" holes, locking the 
nuts in place. In time the bore of the brasses B and C wears to 
oval form. To restore to circular form, one or two liners would be 
removed and the strap replaced. The key driven in would then 
draw the parts closer by the thickness of the liners removed. 

"Drawing (to he Orthographic, not Isometric). — On a sheet 
14"xll" make in ink a working drawing of the front elevation of 
the rod end assembled, viewed in the direction of the arrow. Put 
paper with long dimension horizontal. Put center of bore of 
brasses 4" from left edge of paper and 5" from top edge. No 
sketch, no legend, no dimensions."' 

Problems XIV. 

140. An ordinary brick measures 8"x4"x2-|". Make an ortho- 
graphic drawing and an isometric projection after the manner of 
Fig. 133, Art. 127. Contrast it with the isometric drawing made 
according to Art. 130. 

141. Make the isometric projection of the brick, 8"x4"x2-J", 
turning it through the angles of 45° and 35° 44', as in Pig. 134, 
Art. 129. 

142. From Fig. 135 make a plan and front elevation of the 
strap D. 

143. From Fig. 135 make a plan and front elevation of the stub 
end A. 

144. From Fig. 135 make a plan and front elevation of the 
brass C. 



SET OF DESCRIPTIVE DRAWINGS. 

The following four drawing sheets are designed to be executed in 
the drawing room to illustrate those principles of Descriptive 
Geometry which have the most freouent application in Mechanical 
or Engineering Drawing. 

The paper used should be about 28"x22", the drawing-board of 
the same size, and the blade of the T-square 30". 

To lay out the sheets find the center, approximately, draw center 
lines, and draw three concentric rectangles, measuring 24" x 18", 
22"xl6", and 21"xl5". The outer rectangle is the cutting line 
to which the sheets are to be trimmed. The second one is to be inked 
for the border line. The inner one is described in pencil only as a 
"working line," or line outside of which no part of the actual 
figures should extend. The center lines and other fine lines, in- 
cluding dimensions, may extend beyond the working line. In the 
lower right corner reserve a rectangle 6" X 3", touching the working 
lines, for the legend of the drawing. 

In making the drawings three widths of line are used. 

The actual lines of the figures must be " standard lines " or lines 
not quite one-hundredth of an inch thick. The thin metal erasing 
shield may be used as a gauge for setting the right-line pen, by so 
adjusting the pen that the shield will slowly slip from between the 
nibs, when inserted and allowed to hang vertically. Visihle edges 
are full lines. Hidden edges are broken lines ; the dashes -J" long 
and spaces -j^" long. 

The extra-fine lines are described with the pen adjusted to as fine 
a line as it Avill carry continuously. The axes of projection are 
fine full lines. The dimension lines are long dashes, \" to 1" long, 
with ^" spaces. The center lines are long dashes with fine dots 
between the dashes, or are dash-dot lines. The construction lines 
are long dashes with two dots between, or are dash-dot-dot lines. 
When auxiliar}'- cutting planes are used, one only, together with its 
corresponding projection lines, should be inked in this manner. 
11 



150 Engineering Desgeiptive Geometry 

The extra-heavy lines are about two-hundredths of an inch thick, 
and are for two purposes : for shade lines, if used ; and for paths of 
sections, or lines showing where sections have been taken, as pq. 
Fig. 32. These paths of sections should be formed of dashes 
about ^" long. 

SHEET I: PRISMS AND PYRAMIDS. 

Lay out the sheet and from the center of the sheet plot three ori- 
gins : The first origin 5^" to the left and 4rJ" above the center of the 
sheet ; the second 8" to the right and 2-J" above the center ; and the 
third 4" to the left and 4^" below the center. Pass vertical and 
horizontal lines through these points to act as axes of projection. 

First Origin: Pentagonal Prism and Inclined Plane. 

Describe a pentagonal prism, the axis extending from P (2", 
If", i") to P' (2", If", U"). The top base is a regular pentagon 
inscribed in a circle of 1^" radius, one corner of the pentagon 
being at A (2", ^", |"). Draw three views of the prism. Draw the 
traces of a plane P, perpendicular to V? i^^s trace on V passing 
through the point c (0", 0", 2-|") and making an angle of 60° with, 
the axis of Z. Draw on the side elevation the line of intersection 
of the prism and the plane P. Show the true shape of the polygonal 
line of intersection on an auxiliary plane \], perpendicular to V> 
its traces on V passing through the point (0", 0", 4^"). On U 
show only the section cut by the plane. Draw the development 
of the surface of the prism, with the line of intersection described 
on it. Draw the left edge of the development [representing 
A (2", i", i"),A' (2", i", 2i")] as a vertical line i" to the right of 
the axis of Y, and use the top working edge of the sheet as the top 
line of the development. Omit the pentagonal bases. 

Second Origin: Octagonal Prism and Triangular Prism. 

Describe an octagonal prism, the axis extending from P (2^", 
If" i") to P' (21", If", 41"). The octagonal base is circumscribed 
about a circle of 2^" diameter, one flat side being parallel to the 



Set of Descriptive Drawings 151 

axis of X. Describe a triangular prism, its axis extending from 
Q (0'.'52, If", li") to Q' (3:'98, If", 31"), intersecting PP' at its 
middle point and making an angle of 60° Tvitli it. The base is in 
a plane perpendicular to QQ', and is an equilateral triangle cir- 
cumscribed about a circle of 1" diameter. One corner is at 
J (1", If", 0".38). Draw the H, V, and S projections of the 
prisms and a complete projection on a plane \J, taken perpendicular 
to QQ', and whose trace on V passes through the point (6", 0", 0"). 
Draw the triangular prism as if piercing the octagonal prism. 

Third Origin: Hexagonal Pyramid and Square Prism. 

Describe an hexagonal pyramid, vertex at P (If", 3", i"), center 
of base at P' (If", 2", 3") . The hexagonal base is in a plane parallel 
to H and is circumscribed about a circle 2^" in diameter, one 
corner being at A (If", 0"5G, 3"). Projecting from the sides of the 
pyramid are two portions of a square prism, whose axis is Q (^", 
3", H"), Q' (3i", 2", 2^"). The square base is in a plane parallel 
to S and measures 1" on each edge, and its edges are parallel to the 
axes of Y and Z. Letter the edges GQ' , HH', etc., the point G 
being (i", 1^", If"), E (i", 2^", If"), etc. Draw the object as if 
cut from one solid piece of material, the prism not piercing the 
pyramid. 

The views required are plan, front elevation, and side elevation, 
and also an auxiliary projection on a plane KJ, perpendicular to H, 
The H trace of U makes an angle of 120° with the axis of X at 
the point Z m", 0", 0"). 

Draw also the developments of the surfaces. Place the vertex of 
the developed pyramid at a point ^" to the right and 3^" above the 
origin, and the point A ^" to the right and 0."36 above the origin. 
Mark the line of intersection with the prism on this development. 

Between the side elevation and the legend space, draw the de- 
velopment of the square prism, placing the long edges, GO', HH', 
etc., in a vertical position. Describe the line of intersection on the 
development. Let the edge which has been opened out be GG', and 
let the middle portion of the prism, which does not in reality exist, 
be drawn with construction lines. % 



153 Engineering Descriptive Geometry 

General Directions for Completing the Sheet. 

In inking the sheet show one line of projection for the determi- 
nation of one point on each line of intersection. Shade the figure, 
except the developments. 

In the legend space make the following legend : 

SHEET I. (Block letters 15/32" high.) 

DESCRIPTIVE GEOMETRY. (AH caps 3/16" high.) 

PRISMS AND PYRAMIDS. (All caps 9/32" high.) 

Name (signature). Class. (Caps 1/8" high, lower case 1/12" high.) 

Date. (Caps 1/8" high, lower case 1/12" high.) 



SHEET II: CYLINDERS, ETC. 

Lay out cutting, border, and working lines, and legend space as 
before. 

Plot four points of origin as follows : Eirst origin, 6" to the left 
and 4" above the center of the sheet; second origin, 4f" to the right 
and 4^" above the center ; third origin, 6^" to the left and 3 J" 
below the center; fourth origin, 6^" to the right and 4^" below the 
center. 

First Origin: Intersecting Right Cylinders. 

Draw the three views of two intersecting right cylinders. The 
axis of one is P (3^", 2", ^"), P' (2^", 2", 3^"), and its diameter is 
3". The axis of the other^is Q {V , If", 2"), Q' (4i", If", 2"), and 
its diameter 2f ". Determine the line of intersection in V ^J planes 
parallel to V at distances of f", 1", li", etc. 

Second Origin: Inclined Cylinder and Inclined Plane. 

Draw three views of an inclined circular C3dinder, cut by a plane. 
The axis of the cylinder is P (3.73", If", V), P' (2", If", 3^')- 
The base is a circle, diameter 2-J", in a plane parallel to H. The 
plane cutting the cylinder is perpendicular to \, and its trace in 
V passes through the middle point of PP', and inclines up to the 



Set of Descriptive Drawings 153 

left at an angle of 30° with OX. Plot the intersection in H, V, 
and S and find the true shape of the ellipse by an auxiliary plane 
of projection perpendicular to V through the point (3", 0", 4"). 

Third Origin: Eight Circular Cone and Inclined Plane. 

Draw a right circular cone, vertex at P {2", If", ^"), center of 
base at P' (2", If", 4"), diameter of base 3". The cone is inter- 
sected by a plane perpendicular to S, having its trace in S parallel 
to the extreme right element of the cone and through the point 
(0", 2^", A"). Draw the line of intersection in plan and front 
elevation, and show the true shape of the curve by projection on an 
auxiliary plane KJ perpendicular to S^ its trace passing through 
the point (0", 2^", 0"). 

Fourth Origin: Ogival Point, Vertical Plane and Inclined Plane. 

Let S lie to the right of H and make no n.se of V- The problem 
is to draw two views of a 3^" ogival shell, intersected by two planes. 
The ogival point is generated by revolving 60° of arc of 3y radius 
about an axis perpendicular to H at the point (2", If", 0"). The 
initial position of the generating arc is as follows: The center is 
at D (0", 31", 3i"), one extremity is at B (0", 0", 3^"), and one is at 
P (2", If", 0.46"). The cylindrical body of the shell extends from 
the ogival point to the right in the side elevation, a distance of f ". 
Two planes, T and R, intersect the shell. T is parallel to and 1^" 
from S- R is perpendicular to S? and its S trace passes through 
the origin, and makes angles of 45° with the axis of Y and the 
axis of Z. Draw: The traces of T and P; the side elevation; the 
line of intersection of T with the shell; and, on the plan, the line 
of intersection of R with the shell. 

General Directions for Completing the Sheet. 

In inking the sheet show one cutting plane for the determination 
of each line of intersection, and show clearly how one point is de- 
termined in each view of each figure. Shade the figure except the 
developments. 



154 Engineekixg Descriptive Geometky 

In the legend space make the following legend: 

SHEET II. (Block letters 15/32" high.) 

DESCRIPTIVE GEOMETRY. (All caps 3/16" high.) 

intersections of CYLINDERS, ETC. (AH caps 9/32" high.) 

Name (signature). ClaSS. (Caps 1/8" high, lower case 1/12" high.) 

Date. (Caps l/S" high, lower case 1/12" high.) 

SHEET III: SURFACES OF REVOLUTION. 

Lay ont center lines, cutting, border and working lines, and 
legend space as before. 

Plot five points of origin as follows : Eirst origin, 6f " to the 
left and 3f" above the center of the sheet; second origin, 1^" to 
the right and 6" above the center; third origin, 8f" to the right and 
5f" above the center; fourth origin, 6-|" to the left and 4f" below 
the center ; fifth origin, 7f " to the right of the center of the sheet 
on the horizontal center line. 

First Origin: Sphere and Cylinder. 

Draw a sphere pierced by a right circular cylinder. The center 
of the sphere is at (2", 2", 2"), its diameter 3-|". The axis of the 
cylinder is P (2", li",^"), P' (2", li", 3§"). Its diameter is H"- 
Draw the sphere and cylinder in ff, V and S? and determine the 
line of intersection by passing planes parallel to V at distances of 
r, r, H" and If". " 

Second Origin: Forked End of Connecting-Rod. 

The forked end of a connecting rod has the shape of a surface of 
revolution, faced off at the sides to a width of 1^', as shown in Fig. 
136. The centers a, h, and c are points (2", 1", 0"), (2", 0", f"), 
and (2", 0", 1"). The arc which has d as a center is tangent at its 
ends to the adjacent arc and to the side of the 1" cylinder. 

Determine the continuation of the line of intersection of the 
plane and surface at iv, by passing planes parallel to H at distances 
from n of 2i", 2f ", 2i", 21" and 2|". Draw no side view. 



Set of Descriptive Drawings 



155 



Third Origin: Stub End of Connecting-Rod. 

The stub end of a connecting-rod is a surface of revolution faced 
off at the sides to a width of 1^", and pierced by bore-holes parallel 
to its axis as shown in Fig. 137. Centers are at a (1|", 1", 0"), 
6 (3", r, 0"), c {%", 1", 0"), d (31", 0", If), and e {%", 0", 1^"). 
Determine the continuation of the line of intersection at lu by 
passing planes parallel to H at distances from H of IxV"? ^s''^ 




Fig. 136. 



Fig. 137. 



1:^", and 1§". Draw also the side view and determine the ap- 
pearance of the edge marked u, where the large part of the bore-hole 
intersects the surface of revolution, by means of the same system of 
planes. 

Fourth Origin: Right Circular Cylinder and Cone. 

A right circular cone is pierced by a right circular cylinder, the 
axes intersecting at right angles, as in Fig. 62, Art. 72. The axis 



156 Engineering Descriptive Geometry 

of the cone is P {2i", 2^", i"), P' (2f' , 2^", 2f"). The base, in a 
plane parallel to W, is a circle of 3f" diameter. The axis of the 
cylinder is Q {I", 2f' , If"), Q' (4", 2^', If"), and its diameter is 
H". 

Draw three views of the figures, determining the line of inter- 
section by planes parallel to H. It is best not to pass these planes at 
equal intervals, but through points at equal angles on the base of 
the cylinder. Divide the base of the cylinder in S iii^fo arcs of 30°, 
and in numbering the points let that corresponding to F, in Fig. 62, 
be numbered and let H be numbered 6. Insert intermediate 
points from 1 to 5 on both sides, so that the horizontal planes used 
for the determination of the curve of intersection are seven in 
number, the lowest passing through the point 0, the second through 
the two points 1, the third through the two points 2, etc. Determine 
the curve of intersection by these planes. 

Draw the development of the surface of the cylinder, cutting the 
surface on the element 00' (or FF' in Fig. 62). Place this line of 
the development vertically on the sheet, the point being 1" to the 
left and 7^" below the center of the sheet, and 0' being 1" to the 
left and 3f" below the center of the sheet. 

Draw the development of the surface of the cone Note that the 
radius of the base, the altitude, and the slant height are in the 
ratio of 3 : 4 : 5. To get equally spaced elements on the surface of 
the cone, divide the arc corresponding to BC in f\, Fig. 62, into 
five equal spaces. ISTumber the point B and C 5, and the inter- 
mediate points in series. Since the cone is symmetrical about two 
axes at right angles, one quadrant may represent all four quadrants. 
Put the vertex of the developed surface 3" to the left of the center 
of the sheet and 1" below it, and consider it cut on the line PO or 
PB. Locate the point ^" to the left of the center of the sheet and 
1" below it. Divide the development into four quadrants and then 
divide each quadrant into five parts, numbering the 21 points 
0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0. 

Fifth Origin: Cone and Double Ogival Point. 

In this figure a right circular cone pierces a double ogival point. 
The cone has a vertical axis, PP', the vertex P being at (3", 1^", 



Set of Descriptive Drawings 157 

1"), and P', the center of the base, at (3", l^", 3^"). The base is a 
circle of 2^" diameter lying in a horizontal plane. 

The ogival point has an axis of revolution, Q (|", 1-|", 2"), 
Q' (5f", U", 2"), H" long. The generating line is an arc of 4" 
radius of which QQ' is the chord, and in its initial position the arc 
has its center at (3", IV', o'.'02). Draw three views of the cone 
piercing the double ogival surface, and determine the line of inter- 
section by means of three auxiliary cutting spheres, centered at p, 
the intersection of PP' and QQ'. Use diameters of 2-|", 2^", and 
2y^g-". This curve appears on the U. S. Navy standard 3" valve. 

General Directions for Completing the Sheet. 

In inking the sheet show one cutting plane or sphere for the 
determination of each line of intersection, and show clearly how 
one point is determined in each view of each figure. Shade the 
figures, except the developments. 

In the legend space record the following legend : 

SHEET III. (Block letters 15/32" high.) 

DESCRIPTIVE GEOMETRY. (All caps 3/16" high.) 
INTERSECTIONS OF SURFACES OF ""- •^- 

REVOLUTION, (All caps 9/32" high.) 

Name (signature). ClaSS. (Caps 1/S" high, lower case 1/12" high.) 

Date. (Caps 1/8" high, lower case 1/12" high.) 



SHEET IV: CONES, ANCHOR RING AND HELICOIDAL 
SURFACES. 

Lay out center lines, cutting, border, working lines, and legend 
space as before. 

From the center of the sheet plot origins as follows : First 
origin, 3^" to the left of the center and 3y|" above the center; 
second origin, 5-J" to the right of the center and 3" above the center ; 
third origin, 10^" to the right of the center and 4^" above the center; 
fifth origin, 3" to the right of the center and 6" below the center. 



158 Engineeeing Descriptive Geometry 

First Origin: Intersecting Inclined Cones. 

Draw two intersecting inclined cones. The first cone has its 
vertex at P (1", 1|", i"), and the center of its base at P' {2",{l 
1|", 4f"). The base is a circle of 3|" diameter, l3dng in a plane 
parallel to ff. The second cone has its vertex at Q (5", If", 2" 4:6), 
and the center of its base at Q' (^", 1^", 3^"). The base is a circle 
of 3" diameter lying in a plane parallel to S- Draw plan, front 
elevation, side elevation, and an auxiliary projection on a plane \], 
perpendicular to the line PQ, the trace of U on V? passing through 
the point M (7^", 0", 0"). Determine the line of intersection of the 
cones by auxiliary cutting planes containing the line PQ, and treat 
the problem on the supposition that the cone PP' pierces the cone 

QQ'- 

Second Origin: Helicoidal Surface for Screw Propeller. 

A right vertical cylinder, 1^" in diameter, has for its axis 
P (2V, 2Y, Y), P' m", H", H")- Projecting from the cylinder 
is a line A (3i", 2^", -J"), B (41", 2^", i"). This line, moving 
uniformly along the cylinder, and about it clockwise, describes one 
complete turn of a helicoidal surface of 3" pitch. Draw plan and 
front elevation of the figure. This helicoid is intersected by an 
elliptical cylinder of which the generating line is perpendicular to 
H and the directrix is an ellipse lying in ff, having its major axis 
C (2", 2V, 0"), D (i", 2V, 0"), and minor axis E (l^", 3", 0"), 
F (ly, 2"j 0")- Find the intersection of the two surfaces. Ink in 
full lines only the circular cylinder and the intersection. This 
portion of a helicoidal surface is similar to that which is used for 
the acting surface of the ordinary marine screw propeller, of 3 or 
4 blades. 

Third Origin: Worm Thread Surface. 

A worm shaft is a right cylinder, 1^" in diameter, its axis being 
P (If", If", i"), P' (If", If", 8f").^ A triple right-hand worm 
thread, of the same profile as in Fig. 70, projects from the cylinder 
along the middle 6" of its length. The pitch of the thread is 4|", 



Set of Descriptive Deawiis^gs 159 

so that each thread has more than a complete turn. The outside 
diameter of the worm is 3". Make a complete drawing of the plan 
and front elevation, as in Fig. 70, letting the worm thread begin at 
any point on tlie circumference. 

Fourth Origin: Anchor Ring and Planes. 

An anchor ring, E, is formed by revolving a circle of 1^" diameter, 
lying in a plane parallel to V ^"^^ with its center at A (1-|", 2f", 
^"), about an axis perpendicular to H and piercing H at the point 
B (2f", 2f", 0"). Draw plan, front elevation, right side elevation 
(to the right of H), and left side elevation (to the left of fl) on a 
plane S'? 4|" from S. A plane P, parallel to S at f " from S, cuts 
the ring. Draw the trace of P on f\, and the intersection of P and 
the ring on S- A second plane P', parallel to S at 1-|" distance, 
cuts the ring. Draw the trace P'H and the intersection P'B on S- 
A third plane Q is parallel to V at If" distance from V- Draw the 
trace QH and the intersection QP on V^ A. fourth plane, Q', is 
parallel to V at 2" distance. Draw the trace Q'H and the inter- 
section Q'R on V- An inclined plane T is perpendicular to S and 
S', its trace on S' passing through the point C (4f", 2f", ^"), and 
inclining down to the right at such an angle as to be tangent to the 
projection on S' of the generating circle when its center is at 
D (2f", 1-J", |-"). Draw the trace of T on S'? and the intersection 
TR on f\. Find the true shape of TR by means of an auxiliary 
plane of projection U perpendicular to S', cutting g' in a trace 
parallel to TS' through the point on g' whose coordinates are 
E (4|", 0", li") . 

General Directions for Completing the Sheet. 

Ink the sheet uniform with the preceding sheets, and in the 
legend space record the following legend : 

SHEET IV. (Block letters 15/32" high.) 

DESCRIPTIVE GEOMETRY. (All caps 3/16" high.) 

CONES, ANCHOR RING AND HELICOIDS. (All caps 9/32" high.) 

Name (signature). ClaSS. (Caps 1/8" high, lower case 1/12" high.) 

Date. (Caps 1/8" high, lower case 1/12" high.) 



Short- TITLE Catalogue 

OF THE 

PUBLICATIONS 

OF 

JOHN WILEY & SONS 

New York 
London: CHAPMAN & HALL, Limited 



ARRANGED UNDER SUBJECTS 



Descriptive circulars sent on application. Books marked with an asterisk (*) are 
sold at net prices only. All books are bound in cloth unless otherwise stated. 



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Maire's Modem Pigments and their Vehicles 12mo. S2 00 

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Berg's Buildings and Structures of American Railroads 4to, 

Bi'ooks's Handbook of Street Railroad Location 16mc, mor. 

Butts's Civil Engineer's Field-book 16mc, mor. 

Crandall's Railway and Other Earthwork Tables 8vo, 

Transition Curve 16mo, mor. 

* Crockett's Methods for Earthwork Computations Svo, 

Dredge's History of the Pennsylvania Railroad. (1879) Paper, 

Fisher's Table of Cubic Yards Cardboard, 

Godwin's Railroad Engineers' Field-book and Explorers' Guide. . 16mo, mor. 
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
bankments Svo, 1 00 

Ives and Hilts's Problems in Surveying, Railroad Surveying and Geodesy 

Ifimo, mor. 1 50 

Molitor and Beard's Manual for Resident Engineers ..... 16mo, 1 00 

Nagle's Field Manual for Railroad Engineers 16mo, mor. 3 00 

* Orrock's Railroad Structures and Estimates - Svo, 3 00 

Philbrick's Field Manual for Engineers 16mo, mor. 3 00 

Raymond's Railroad Engineering. 3 volumes. 

Vol. I. Railroad Field Geometry. (In Preparation.) 

Vol. II. Elements of Railroad Engineering Svo, 3 50 

Vol. III. Railroad Engineer's Field Book. (In Preparation.) 

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Roberts' Track Formulas and Tables. (In Press.) 

Searles's Field Engineering IGmo, mor. $3 00 

Railroad Spiral 16mo, mor. 1 50 

Taylor's Prismoidal Formulae and Earthwork 8vo, 1 50 

* Trautwine's Field Practice of Laying Out Circular Curves for Railroads. 

12mo, mor. 2 50 

* Method of Calculating the Cubic Contents of Excavations and Em- 
bankments by the Aid of Diagrams 8vo, 2 00 

Webb's Economics of Railroad Construction Large 12mo, 2 50 

Railroad Construction 16mo, mor. 5 00 

Wellington's Economic Theory of the Location of Railways Large 12mo, 5 00 

Wilson's Elements of Railroad-Track and Construction 12mo, 2 00 



DRAWING. 



Barr's Kinematics of Machinery 8vo 

* Bartlett's Mechanical Drawing Svo 

* " " " Abridged Ed Svo 

Coolidge's Manual of Drawing Svo, paper 

CooHdge and Freeman's Elements of General Drafting for Mechanical Engi 

neers Oblong 4to 

Durley's Kinematics of Machines Svo 

Emch's Introduction to Projective Geometry and its Application Svo 

French and Ives' Stereotomy Svo 

Hill's Text-book on Shades and Shadows, and Perspective Svo 

Jamison's Advanced Mechanical Drawing Svo 

Elements of Mechanical Drawing Svo 

Jones's Machine Design: 

Part I. Kinematics of Machinery Svo 

Part II. Form, Strength, and Proportions of Parts Svo 

* Kimball and Barr's Machine Design Svo 

MacCord's Elements of Descriptive Geometry Svo 

Kinematics ; or. Practical Mechanism Svo 

Mechanical Drawing 4to 

Velocity Diagrams Svo 

McLeod's Descriptive Geometry Large 12mo 

''■ Mahan's Descriptive Geometry and Stone-cutting Svo 

Industrial Drawing. (Thompson.) • Svo 

Moyer's Descriptive Geometry Svo 

Reed's Topographical Drawing and Sketching 4to 

Reid's Course in Mechanical Drawing , Svo 

Text-book of Mechanical Drawing and Elementary Machine Design. .Svo 

Robinson's Principles of Mechanism Svo 

Schwamb and Merrill's Elements of Mechanism Svo 

Smith (A. W.) and Marx's Machine Design Svo, 

Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo 

* Titsworth's Elements of Mechanical Drawing Oblong Svo 

Warren's Drafting Instruments and Operations 12mo 

Elements of Descriptive Geometry, Shadows, and Perspective Svo 

Elements of Machine Construction and Drawing Svo, 

Elements of Plane and Solid Free-hand Geometrical Drawing. . . . 12mo 

General Problems of Shades and Shadows Svo 

Manual of Elementary Problems in the Linear Perspective of Forms and 

Shadow 12mo 

Manual of Elementary Projection Drawing 12mo 

Plane Problems in Elementary Geometry 12mo 

Weisbach's Kinematics and Power of Transmission. (Hermann and 
Klein.) Svo 

Wilson's (H. M.) Topographic Surveying Svo 

* Wilson's (V. T.) Descriptive Geometry Svo 

Free-hand Lettering Svo 

Free-hand Perspective Svo, 

Woolf's Elementary Course in Descriptive Geometry Large Svo 

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ELECTRICITY AND PHYSICS. 

* Abegg's Theory of Electrolytic Dissociation, (von Ende.) 12mo, 

Andrews's Hand-book for Street Railway Engineering 3X5 inches, mor. 

Anthony and Brackett's Text-book of Physics. (Magie.) ... .Large 12mo, 
Anthony and Ball's Lecture-notes on the Theory of Electrical Measure- 
ments 12mo, 

Benjamin's History of Electricity 8vo, 

Voltaic Cell 8vo, 

Betts's Lead Refining and Electrolysis 8vo, 

Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, 

* CoUins's Manual of Wireless Telegraphy and Telephony 12mo, 

Crehore and Squier's Polarizing Photo-chronograph 8vo, 

* Danneel's Electrochemistry. (Merriam.) 12mo, 

Dawson's "Engineering" and Electric Traction Pocket-book. . . . 16mo, mor. 
Dolezalek's Theory of the Lead Accumulator (Storage Battery), (von Ende.) 

12mo, 

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 

Flather's Dynamometers, and the Measurement of Power 12mo, 

* Getman's Introduction to Physical Science 12mo, 

Gilbert's De Magnete. (Mottelay ) 8vo, 

* Hanchett's Alternating Currents 12mo, 

Hering's Ready Reference Tables (Conversion Factors) 16mo, mor. 

* Hobart and EUis's High-speed Dynamo Electric Machinery 8vo, 

Holman's Precision of Measurements 8vo, 

Telescopic Mirror-scale Method, Adjustments, and Tests.. . .Large 8vo, 

* KarapetofE's Experimental Electrical Engineering 8vo, 

Kinzbrunner's Testing of Continuous-current Machines 8vo, 

Landauer's Spectrum Analysis. (Tingle.) 8vo, 

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess. )12mo, 
Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 

* Lyndon's Development and Electrical Distribution of Water Power. .8vo, 

* Lyons's Treatise on Electromagnetic Phenomena. Vols, I .and II. 8vo, each, 

* Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 

Morgan's Outline of the Theory of Solution and its Results 12mo, 

* Physical Chemistry for Electrical Engineers 12mo, 

* Norris's Introduction to the Study of Electrical Engineering 8vo, 

Norris and Dennison's Course of Problems on the Electrical Characteristics of 

Circuits and Machines. (In Press.) 

* Parshall and Hobart's Electric Machine Design 4to, half mor, 12 50 

Reagan's Locomotives: Simple, Compound, and Electric. New Edition. 

Large 12mo, 3 50 

* Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbrunner.) . .8vo, 2 00 

Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50 

Schapper's Laboratory Guide for Students in Physical Chemistry 12mo, 1 00 

* Tillman's Elementary Lessons in Heat 8vo, 1 50 

Tory and Pitcher's Manual of Laboratory Physics Large 12mo, 2 00 

Ulke's Modem Electrolytic Copper Refining 8vo, 3 00 



LAW. 

* Brennan's Hand-book of Useful Legal Information for Business Men. 

16mo, mor. 5 00 

* Davis's Elements of Law 8vo, 2 50 

* Treatise on the Military Law of United States 8vo, 7 00 

* Dudley's Military Law and the Procedure of Courts-martial. .Large 12mo, 2 50 

Manual for Courts-martial 16mo, mor. 1 50 

Wait's Engineering and Architectural Jurisprudence 8vo, 6 00 

Sheep, 6 50 

Law of Contracts 8vo, 3 00 

Law of Operations Preliminary to Construction in Engineering and 

Architectvire 8vo, 5 00 

Sheep, 5 5Q 
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MATHEMATICS. 

"Baker's Elliptic Functions 8vo, $1 50 

Briggs's Elements of Plane Analytic Geometry. (Bocher.) 12mo, 1 00 

* Buchanan's Plane and Spherical Trigonometry 8vo, 1 00 

Byerley's Harmonic Functions 8vo, 1 00 

Chandler's Elements of the Infinitesimal Calculus 12mo, 2 00 

* Coffin's Vector Analysis 12mo, 2 50 

Compton's Manual of Logarithmic Computations 12mo, 1 50 

* Dickson's College Algebra Large 12mo, 1 50 

* Introduction to the Theory of Algebraic Equations Large 12mo, 1 25 

lEmch's Introduction to Projective Geometry and its Application 8vo, 2 50 

Fiske's Functions of a Complex Variable 8vo, 1 00 

iHalsted's Elementary Synthetic Geometry 8vo, 1 50 

Elements of Geometry 8vo, 1 75 

* Rational Geometry 12mo, 1 50 

Synthetic Projective Geometry 8vo, 1 00 

"Hancock's Lectures on the Theory of Elliptic Functions. (In Press.) 

Hyde's Grassmann's Space Analysis 8vo, 1 00 

'* Johnson's (J. B.) Three-place Logarithmic Tables : Vest-pocket size, paper, 15 

* 100 copies, 5 00 

* Mounted on heavy cardboard, 8 X 10 inches, 25 

* 10 copies, 2 00 
Johnson's (W. W.) Abridged Editions of Differential and Integral Calculus. 

Large 12mo, 1 vol. 2 50 

Curve Tracing in Cartesian Co-ordinates 12mo, 1 00 

Differential Equations 8vo, 1 00 

Elementary Treatise on Differential Calculus Large 12mo, 1 50 

Elementary Treatise on the Integral Calculus .Large 12mo, 1 50 

* Theoretical Mechanics 12mo, 3 00 

Theory of Errors and the Method of Least Squares 12mo, 1 50 

Treatise on Differential Calculus Large 12mo, 3 00 

Treatise on the Integral Calculus Large 12mo, 3 00 

Treatise on Ordinary and Partial Differential Equations. . .Large 12mo, 3 50 

TCarapetoff's Engineering Applications of Higher Mathematics. (In Preparation.) 

X,aplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . 12mo, 2 00 

* Ludlow and Bass's Elements of Trigonometry and Logarithmic and Other 

Tables 8vo, 3 00 

* Trigonometry and Tables published separately Each, 2 00 

* Ludlow's Logarithmic and Trigonometric Tables 8vo, 1 00 

Macfarlane's Vector Analysis and Quaternions 8vo, 1 00 

McMahon's Hyperbolic Functions 8vo, 1 00 

Manning's Irrational Numbers and their Representation by Sequences and 

Series 12mo, 1 25 

Mathematical Monographs. Edited by Mansfield Merriman and Robert 

S. Woodward Octavo, each 1 00 

No. 1. History of Modern Mathematics, by David Eugene Smith. 
No. 2. Synthetic Projective Geometry, by George Bruce Halsted. 
No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper- 
bolic Functions, by James McMahon. No. 5. Harmonic Func- 
tions, by Vvllliam E. Byerly. No. 6. Grassmann's Space Analysis, 
by Edward W. Hyde. No. 7. Probability and Theory of Errors, 
by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, 
by Alexander Macfarlane. No. 9. Differential Equations, by 
William Woolsey Johnson. No. 10. The Solution of Equations, 
by Mansfield Merriman. No. 11. Functions of a Complex Variable, 
by Thomas S. Fiske. 

TVIaurer's Technical Mechanics 8vo, 4 00 

Merriman's Method of Least Squares 8vo, 2 00 

Solution of Equations 8vo, 1 00 

Rice and Johnson's Differential and Integral Calculus. 2 vols, in one. 

Large 12mo, 1 50 

Elementary Treatise on the Differential Calculus Large 12mo, 3 00 

Smith's History of Modern Mathematics 8vo, 1 00 

■* Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One 

Variable 8vo, 2 00 

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* Waterbury's Vest Pocket Hand-book of Mathematics for Engineers. 

2f X5f inches, mor. $1 00 

* Enlarged Edition, Including Tables mor. 1 50 

Weld's Determinants 8vo, 1 00 

Wood's Elements of Co-ordinate Geometry 8vo, 2 00 

Woodward's Probability and Theory of Errors 8vo, 1 00 



MECHANICAL ENGINEERING. 



MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS. 



Bacon's Forge Practice 12mo, 

Baldwin's Steam Heating for Buildings 12mo, 

Barr's Kinematics of Machinery 8vo, 

* Bartlett's Mechanical Drawing 8vo, 

* " " " Abridged Ed 8vo, 

* Burr's Ancient and Modem Engineering and the Isthmian Canal 8vo, 

Carpenter's Experimental Engineering 8vo, 

Heating and Ventilating Buildings 8vo, 

* Clerk's The Gas, Petrol and Oil Engine 8vo, 

Compton's First Lessons in Metal Working 12mo, 

Compton and De Groodt's Speed Lathe 12mo, 

Coolidge's Manual of Drawing 8vo, paper, 

Coolidge and Freeman's Elements of General Drafting for Mechanical En- 
gineers Oblong 4to, 

Cromwell's Treatise on Belts and Pulleys 12mo, 

Treatise on Toothed Gearing 12mo, 

Dingey's Machinery Pattern Making 12mo, 

Durley's Kinematics of Machines 8vo, 

Flanders's Gear-cutting Machinery ^ Large 12mo, 

Flather's Dynamometers and the Measurement of Power 12mo, 

Rope Driving 12mo, 

Gill's Gas and Fuel Analysis for Engineers 12mo, 

Goss's Locomotive Sparks 8vo, 

Greene's Pumping Machinery. (In Preparation.) 

Hering's Ready Reference Tables (Conversion Factors) 16mo, mor. 

* Hobart and Ellis's High Speed Dynamo Electric Machinery 8vo 

Hutton's Gas Engine 8vo 

Jamison's Advanced Mechanical Drawing 8vo 

Elements of Mechanical Drawing 8vo 

Jones's Gas Engine 8vo 

Machine Design: 

Part I. Kinematics of Machinery 8vo 

Part II. Form, Strength, and Proportions of Parts 8vo 

Kent's Mechanical Engineer's Pocket-Book 16mo, mor 

Kerr's Power and Power Transmission 8vo 

* Kimball and Barr's Machine Design 8vo 

* Levin's Gas Engine 8vo 

Leonard's Machine Shop Tools and Methods 8vo 

* Lorenz's Modem Refrigerating Machinery. (Pope, Haven, and Dean). .8vo 
MacCord's Kinematics; or. Practical Mechanism 8vo 

Mechanical Drawing 4to 

Velocity Diagrams 8vo 

MacFarland's Standard Reduction Factors for Gases 8vo 

Mahan's Industrial Drawing. (Thompson.) 8vo 

Mehrtens's Gas Engine Theory and Design Large 12mo 

Oberg's Handbook of Small Tools Large 12mo 

* Parshall and Hobart's Electric Machine Design. Small dto, half leather, 

Peele's Compressed Air Plant for Mines 8vo 

Poole's Calorific Power of Fuels 8vo 

* Porter's Engineering Reminiscences, 1855 to 1882 8vo 

Reid's Course in Mechanical Drawing 8vo 

Text-book of Mechanical Drawing and Elementary Machine Design. 8vO; 

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Richards's Compressed Air 12mo, $1 50 

Robinson's Principles of Mechanism 8vo, 

Schwamb and Merrill's Elements of Mechanism 8vo, 

Smith (A. W.) and Marx's Machine Design 8vo, 

Smith's (O.) Press-working of Metals 8vo, 

Sorel's Carbureting and Combustion in Alcohol Engines. (Woodward and 

Preston.) Large 12mo, 

Stone's Practical Testing of Gas and Gas Meters 8vo, 

Thurston's Animal as a Machine and Prime Motor, and the Laws of Energetics. 

12mo, 
Treatise on Friction and Lost Work in Machinery and Mill 'Work. . .8vo, 

* Tillson's Complete Automobile Instructor 16mo, 

* Titsworth's Elements of Mechanical Drawing Oblong 8vo, 

Warren's Elements of Machine Construction and Drawing 8vo, 

* Waterbury's Vest Pocket Hand-book of Mathematics for Engineers. 

21X51 inches, mor. 

* Enlarged Edition, Includin-g Tables mor. 

Weisbach's Kinematics and the Power of Transmission. (Herrmann — 

Klein.) 8vo, 

Machinery of Transmission and Governors. (Hermann — Klein.). .8vo, 
Wood's Turbines 8vo, 



MATERIALS OF ENGINEERING. 

* Bovey's Strength of Materials and Theory of Structures Svo, 7 50 

Burr's Elasticity and Resistance of the Materials of Engineering Svo, 7 50 

Church's Mechanics of Engineering 8vo, 6 00 

* Greene's Structural Mechanics 8vo, 2 50 

* HoUey's Lead and Zinc Pigments Large 12mo 3 00 

Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes. 

Large 12mo, 2 50 
Johnson's (C. M.) Rapid Methods for the Chemical Analysis of Special 

Steels, Steel-Making Alloys and Graphite Large 12mo, 3 00 

Johnson's (J. B.) Materials of Construction Svo, 6 00 

Keep's Cast Iron Svo, 2 50 

Lanza's Applied Mechanics Svo, 7 50 

Maire's Modem Pigments and their Vehicles 12mo, 2 00 

Maurer's Technical Mechanics Svo, 4 00 

Merriman's Mechanics of Materials Svo, 5 00 

* Strength of Materials 12mo, 1 00 

Metcalf 's Steel. A Manual for Steel-users 12mo, 2 00 

Sabin's Industrial and Artistic Technology of Paint and Varnish Svo, 3 00 

Smith's ((A. W.) Materials of Machines 12mo, 1 00 

* Smith's (H. E.) Strength of Material 12mo, 1 25 

Thurston's Materials of Engineering 3 vols., Svo, S 00 

Part I. Non-metallic Materials of Engineering, Svo, 2 00 

Part II. Iron and Steel Svo, 3 50 

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents Svo, 2 50 

Wood's (De V.) Elements of Analytical Mechanics Svo, 3 00 

Treatise pn the Resistance of Materials and an Appendix on the 

Preservation of Timber Svo, 2 00 

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and 

Steel Svo. 4 00 



STEAM-ENGINES AND BOILERS. 

Berry's Temperature-entropy Diagram 12mo, 2 00 

Camot's Reflections on the Motive Power of Heat. (Thurston.) 12mo, 1 50 

Chase's Art of Pattern Making 12mo, 2 50 

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Creighton's Steam-engine and other Heat Motors 8vo, 

Dawson's "Engineering" and Electric Traction Pocket-book. .. . 16mo, mor, 

* Gebhardt's Steam Power Plant Engineering 8vo 

Goss's Locomotive Performance 8vo 

Hemenway's Indicator Practice and Steam-engine Economy 12mo 

Hutton's Heat and Heat-engines 8vo 

Mechanical Engineering of Power Plants 8vo 

Kent's Steam boiler Economy 8vo 

Kneass's Practice and Theory of the Injector 8vo 

MacCord's Slide-valves 8vo 

Meyer's Modem Locomotive Construction 4to 

Moyer's Steam Turbine 8vo 

Peabody's Manual of the Steam-engine Indicator 12mo 

Tables of the Properties of Steam and Other Vapors and Temperature- 
Entropy Table 8vo, 

Thermodynamics of the Steam-engine and Other Heat-engines. . . .8vo 

Valve-gears for Steam-engines 8vo 

Peabody and Miller's Steam-boilers 8vo 

Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors, 

(Osterberg.) 12mo 

Reagan's Locomotives : Simple, Compound, and Electric. New Edition. 

Large 12mo 

Sinclair's Locomotive Engine Running and Management 12mo 

Smart's Handbook of Engineering Laboratory Practice 12mo 

Snow's Steam-boiler Practice 8vo 

Spangler's Notes on Thermodynamics 12mo 

Valve-gears 8vo 

Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo 

Thomas's Steam-turbines 8vo 

Thurston's Handbook of Engine and Boiler Trials, and the Use of the Indi- 
cator and the Prony Brake 8vo 

Handy Tables 8vg 

Manual of Steam-boilers, their Designs, Construction, and Operation 8vo 

Manual of the Steam-engine 2 vols., 8vo 

Part I. History, Structure, and Theory 8vo 

Part II. Design, Construction, and Operation 8vo 

Wehrenfennig's Analysis and Softening of Boiler Feed-water. (Patterson.) 

8vo 

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo 

Whitham's Steam-engine Design 8vo 

Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo 



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MECHANICS PURE AND APPLIED. 



Church's Mechanics of Engineering 8vo, 

Notes and Examples in Mechanics 8vo, 

Dana's Text-book of Elementary Mechanics for Colleges and Schools .12mo, 
Du Bois's Elementary Principles of Mechanics: 

Vol. I. Kinematics 8vo, 

Vol. II. Statics. 8vo, 

Mechanics of Engineering. Vol. I Small 4to, 

Vol. II Small 4to, 

* Greene's Structural Mechanics 8vo, 

Hartmann's Elementary Mechanics for Engineering Students. (In Press.) 
James's Kinematics of a Point and the Rational Mechanics of a Particle. 

Large 12mo, 

* Johnson's (W. W.) Theoretical Mechanics 12mo, 

Lanza's Applied Mechanics 8vo, 

* Martin's Text Book on Mechanics, Vol. I, Statics 12mo, 

* Vol. II, Kinematics and Kinetics. 12mo, 
Maurer's Technical Mechanics 8vo, 

* Merriman's Elements of Mechanics 12mo, 

Mechanics of Materials 8vo, 

* Michie's Elements of Analytical Mechanics 8vo, 

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Robinson's Principles of Mechanism 8vo, $3 00 

Sanborn's Mechanics Problems Large 12mo, 1 50 

Schwamb and Merrill's Elements of Mechanism 8vo, 3 00- 

Wood's Elements of Analytical Mechanics 8vo, 3 00 

Principles of Elementary Mechanics 12mo, 1 25. 

MEDICAL. 

* Abderhalden's Physiological Chemistry in Thirty Lectures. (Hall and 

Defren.) Svo, 

von Behring's Suppression of Tuberculosis. (Bolduan.) 12mo, 

Bolduan's Immune Sera 12mo, 

Bordet's Studies in Immunity. (Gay.) Svo, 

Chapin's The Sources and Modes of Infection. (In Press.) 
Davenport's Statistical Methods with Special Reference to Biological Varia- 
tions 16mo, mor. 

Ehrlich's Collected Studies on Immunity. (Bolduan.) Svo, 

* Fischer's Physiology of Alimentation Large 12mo, 

de Fursac's Manual of Psychiatry. (RosanoflE and Collins.).. . .Large 12mo, 

Hammarsten's Text-book on Physiological Chemistry. (Mandel.) Svo, 

Jackson's Directions for Laboratory Work in Physiological Chemistry. .Svo, 

Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) 12mo, 

Mandel's Hand-book for the Bio-Chemical Laboratory 12mo, 

* Nelson's Analysis of Drugs and Medicines 12mo. 

* Pauli's Physical Chemistry in the Service of Medicine. (Fischer.) ..12mo, 

* Pozzi-Escot's Toxins and Venoms and their Antibodies. (Cohn.). . 12mo, 

Rostoski's Serum Diagnosis. (Bolduan.) 12mo, 

Ruddiman's Incompatibilities in Prescriptions Svo, 

Whys in Pharmacy 12mo, 

Salkowski's Physiological and Pathological Chemistry. (Orndorff.) .. ..Svo, 

* Satterlee's Outlines of Human Embryology 12mo, 

Smith's Lecture Notes on Chemistry for Dental Students Svo, 

* Whipple's Tyhpoid Fever Large 12mo, 

* Woodhull's Military Hygiene for Officers of the Line Large 12mo, 

* Personal Hygiene 12mo, 

Worcester and Atkinson's Small Hospitals Establishment and Maintenance, 
and Suggestions for Hospital Architecture, with Plans for a Small 
Hospital 12mo, 1 25= 

METALLURGY. 

Betts's Lead Refining by Electrolysis Svo, 4 GO' 

Holland's Encyclopedia of Founding and Dictionary of Foundry Terms used 

in the Practice of Moulding 12mo, 

Iron Founder 12mo, 

" " Supplement 12mo, 

Douglas's Untechnical Addresses on Technical Subjects 12mo, 

Goesel's Minerals and Metals: A Reference Book 16mo, mor. 

* Iles's Lead-smelting 12mo, 

Johnson's Rapid Methods for the Chemical Analysis of Special Steels, 

Steel-making Alloys and Graphite Large 12mo, 

Keep's Cast Iron Svo, 

Le Chatelier's, High- temperature Measurements. (Boudouard — Burgess.) 

12mo, 

Metcalf's Steel. A Manual for Steel-users 12mo, 

Minet's Production of Aluminum and its Industrial Use. (Waldo.). . 12mo, 

* Ruer's Elements of Metallography. (Mathewson.) Svo, 

Smith's Materials of Machines 12mo, 

Tate and Stone's Foundry Practice 12mo, 

Thurston's Materials of Engineering. In Three Parts Svo, 

Part I. Non-metallic Materials of Engineering, see Civil Engineering, 

page 9. 

Part II. Iron and Steel Svo, 

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their 

Constituents Svo, 

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00' 


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4 


00' 


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25 



Ulke's Modern Electrolytic Copper Refining 8vo, $3 00- 

West's American Foundry Practice 12mo, i" 50 

Moulders' Text Book 12mo. 2 50. 



MINERALOGY. 

Baskerville's Chemical Elements. (In Preparation.) 

* Browning's Introduction to the Rarer Elements 8vo, 

Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 

Butler's Pocket Hand-book of Minerals 16mo, mor. 

Chester's Catalogue of Minerals 8vo, paper, 

Cloth, 

if Crane's Gold and Silver 8vo, 

Dana's First Appendix to Dana's New "System of Mineralogy". .Large 8vo, 
Dana's Second Appendix to Dana's New " System of Mineralogy." 

Large 8vo, 

Manual of Mineralogy and Petrography 12mo, 

Minerals and How to Study Them 12mo, 

System of Mineralogy Large 8vo, half leather. 

Text-book of Mineralogy 8vo, 

Douglas's Un technical Addresses on Technical Subjects 12mo, 

Eakle's Mineral Tables 8vo, 

Eckel's Stone and Clay Products Used in Engineering. (In Preparation.) 

Goesel's Minerals and Metals: A Reference Book. . 16mo, mor. 3 00 

Groth's The Optical Properties of Crystals. (Jackson.) (In Press.) 

Groth's Introduction to Chemical Crystallography (Marshall). ...... . 12mo, 1 25 

* Hayes's Handbook for Field Geologists 16mo, mor. 1 50 

Iddings's Igneous Rocks 8vo, 5 00 

Rock Minerals 8vo, 5 00 

Johannsen's Determination of Rock-forming Minerals in Thin Sections. 8vo, 

With Thumb Index 5 00 

* Martin's Laboratory Guide to Qualitative Analysis with the Blow- 

pipe , 12mo, 60 

Merrill's Non-metallic Minerals; Their Occurrence and Uses 8vo, 4 00 

Stones for Building and Decoration 8vo, 5 00 

* Peniield's Notes on Determinative Mineralogy and Record of Mineral Tests. 

8vo, paper, 50 
Tables of Minerals, Including the Use of Minerals and Statistics of 

Domestic Production 8vo, 1 00 

* Pirsson's Rocks and Rock Minerals 12mo, 2 50 

* Richards's Synopsis of Mineral Characters 12mo, mor. 1 25 

* Ries's Clays: Their Occurrence, Properties and Uses 8vo, 5 00 

* Ries and Leighton's History of the Clay-working Industry of the United 

States 8vo, 2 50 

* Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00 

Washington's Manual of the Chemical Analysis of Rocks 8vo, 2 00 



MINING. 

* Beard's Mine Gases and Explosions Large 12mo, 3 00 • 

* Crane's Gold and Silver 8vo, 5 00 

* Index of Mining Engineering Literature 8vo, 4 00 

* 8vo, mor. 5 00 

Ore Mining Methods. (In Press.) 

Douglas's Untechnical Addresses on Technical Subjects 12mo, 1 00 

Eissler's Modern High Explosives 8vo, 4 00 

Goesel's Minerals and Metals: A Reference Book 16mo, mor. 3 00 

Ihlseng's Manual of Mining 8vo, 5 00 

* Iles's Lead Smelting 12mo, 2 50 

Peele's Compressed Air Plant for Mines 8vo, 3 00 

Riemer's Shaft Sinking Under Difficult Conditions. (Corning and Peele.)8vo, 3 00 

* Weaver's Military Explosives 8vo, 3 00 

Wilson's Hydraulic and Placer Mining. 2d edition, rewritten 12mo, 2 50 

Treatise on Practical and Theoretical Mine Ventilation 12mo, 1 25.. 

17 



SANITARY SCIENCE. 

Association of State and National Food and Dairy Departments, Hartford 

Meeting, 1906 8vo, $3 00 

Jamestown Meeting, 1907 8vo, 3 00 

* Bashore's Outlines of Practical Sanitation 12mo, 1 25 

Sanitation of a Country House 12mo, 1 00 

Sanitation of Recreation Camps and Parks 12mo, 1 00 

Chapin's The Sources and Modes of Infection. (In Press.) 

Polwell's Sewerage. (Designing, Construction, and Maintenance.) 8vo, 3 00 

Water-supply Engineering Svo, 4 00 

Fowler's Sewage Works Analyses 12mo, 2 00 

Fuertes's Water-filtration Works 12mo, 2 50 

Water and Public Health 12mo, 1 50 

Gerhard's Guide to Sanitary Inspections 12mo, 1 50 

* Modern Baths and Bath Houses Svo, 3 00 

Sanitation of Public Buildings 12mo, 1 50 

* The Water Supply, Sewerage, and Plumbing of Modern City Buildings. 

8vo,- 4 00 

Hazen's Clean Water and How to Get It Large 12mo, 1 50 

Filtration of Public Water-supplies 8vo, 3 00 

Kinnicut, Winslow and Pratt's Purification of Sewage. (In Preparation.) 
Leach's Inspection and Analysis of Food with Special Reference to State 

Control 8vo, 7 50 

Mason's Examination of Water. (Chemical and Bacteriological) 12mo, 1 25 

Water-supply. (Considered principally from a Sanitary Standpoint). 

8vo, 4 00 

* Merriman's Elements of Sanitary Engineering Svo, 2 00 

Ogden's Sewer Construction Svo, 3 00 

Sewer Design 12mo, 2 00 

Parsons's Disposal of Municipal Refuse Svo, 2 00 

Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- 
ence to Sanitary Water Analysis 12mo, 1 60 

* Price's Handbook on Sanitation 12mo, 1 50 

Richards's Cost of Cleanness 12mo, 1 00 

Cost of Food. A Study in Dietaries 12mo, 1 00 

Cost of Living as Modified by Sanitary Science 12mo, 1 00 

Cost of Shelter 12mo, 1 00 

* Richards and Williams's Dietary Computer Svo, 1 50 

Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
point Svo, 2 00 

* Richey's Plumbers', Steam-fitters', and Tinners' Edition (Building 

Mechanics' Ready Reference Series) 16mo, mor. 1 50 

Rideal's Disinfection and the Preservation of Food Svo, 4 00 

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Tumeaure and Russell's Public Water-supplies Svo, 5 00 

Venable's Garbage Crematories in America Svo, 2 00 

Method and Devices for Bacterial Treatment of Sewage Svo, 3 00 

Ward and Whipple's Freshwater Biology. (In Press.) 

Whipple's Microscopy of Drinking-water Svo, 3 50 

* Typhoid Fever Large 12mo, 3 00 

Value of Pure Water Large 12mo, 1 00 

Winslow's Systematic Relationship of the Coccaceas Large 12mo, 2 50 



MISCELLANEOUS. 

Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 

International Congress of Geologists Large Svo. 1 50 

Ferrel's Pooular Treatise on the Winds Svo, 4 00 

Fitzgerald's Boston Machinist ISmo, 1 00 

Gannett's Statistical Abstract of the World 24mo. 75 

Haines's American Railway Management 12mo, 2 50 

Hanausek's The Microscopy of Technical Products. (Winton) Svo, 5 00 

18 



Jacobs's Betterment Briefs. A Collection of Published Papers on Or- 
ganized Industrial Eificiency 8vo, $3 50 

Metcalfe's Cost of Manufactures, and the Administration of Workshops.. 8 vo, 5 00 

Putnam's Nautical Charts Svo, 2 00 

Ricketts's History of Rensselaer Polytechnic Institute 1824-1894. 

Large 12mo, 3 00 

Rotherham's Emphasised New Testament Large Svo, 2 00 

Rust's Ex-Meridian Altitude, Azimuth and Star-finding Tables Svo, 5 00 

Standage's Decoration of Wood, Glass, Metal, etc 12mo, 2 00 

Thome's Structural and Physiological Botany. (Bennett) 16mo, 2 25 

Westermaier's Compendium of General Botany. (Schneider) Svo, " 00 

Winslow's Elements of Applied Microscopy 12mo, 1 50 



HEBREW AND CHALDEE TEXT-BOOOKS. 

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 

(Tregelles.) Small 4to, half mor, 5 00 

Green's Elementary Hebrew Grammar 12mo, 1 25 



19 



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